Research ArticleTerminal-Dependent Statistical Inference forthe FBSDEs Models

Yunquan Song12

1 China University of Petroleum Qingdao 266580 China2 Shandong University Qilu Securities Institute for Financial Studies Shandong University Jinan 250100 China

Correspondence should be addressed to Yunquan Song math1212163com

Received 12 March 2014 Accepted 27 May 2014 Published 25 June 2014

Academic Editor Guangchen Wang

Copyright copy 2014 Yunquan SongThis is an open access article distributed under theCreativeCommonsAttributionLicense whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The original stochastic differential equations (OSDEs) and forward-backward stochastic differential equations (FBSDEs) are oftenused to model complex dynamic process that arise in financial ecological and many other areas The main difference betweenOSDEs and FBSDEs is that the latter is designed to depend on a terminal condition which is a key factor in some financial andecological circumstances It is interesting but challenging to estimate FBSDEparameters fromnoisy data and the terminal conditionHowever to the best of our knowledge the terminal-dependent statistical inference for such a model has not been explored inthe existing literature We proposed a nonparametric terminal control variables estimation method to address this problem Thereason why we use the terminal control variables is that the newly proposed inference procedures inherit the terminal-dependentcharacteristicThrough this new proposed method the estimators of the functional coefficients of the FBSDEs model are obtainedThe asymptotic properties of the estimators are also discussed Simulation studies show that the proposed method gives satisfyingestimates for the FBSDE parameters from noisy data and the terminal condition A simulation is performed to test the feasibilityof our method

1 Introduction

Since 1973 when the worldrsquos first options exchange openedin Chicago a large number of new financial products havebeen introduced to meet the customerrsquos demands from thederivative markets In the same year Black and Scholes [1]provided their celebrated formula for option pricing andMerton [2] proposed a general equilibriummodel for securityprices Since then modern finance has adopted stochasticdifferential equations as its basic instruments for portfoliomanagement asset pricing risk management and so onAmong these models the backward stochastic differentialequations (BSDEs for short) are a desirable choice for hedgingand pricing an option Its general form is as follows

119889119884119904= minus119892 (119904 119884

119904 119885

119904) 119889119904 + 119885

119904119889119861

119904

119884119879= 120585 119904 isin [119905 119879]

(1)

where 119892 is the generator 119861119905is a Brownian motion and 120585 is a

R-valued Borel function as the terminal condition Usually

the terminal condition is designed as a random variable withgiven distribution If 119892 meets certain conditions the BSDEhas a unique solution

In terms of the backward equation within a completemarket it serves to characterize the dynamic value of repli-cating portfolio 119884

119904with a final wealth 120585 and a special quantity

119885119904that depends on the hedging portfolio In particular while

the generator consists of diffusion process the correspondingequation is proved to be a forward-backward stochasticdifferential equation (FBSDE) which can be expressed as

119889119884119904= minus119892 (119904 119883

119904 119884

119904 119885

119904) 119889119904 + 119885

119904119889119861

119904 119884

119879= 120585 (2)

where 119883119904satisfies the following ordinary stochastic differen-

tial equation (OSDE)

119889119883119904= 120583 (119904 119883

119904) 119889119905 + 120590 (119904 119883

119904) 119889119861

119904 119904 isin [119905 119879] (3)

Compared to the OSDE that contains an initial condition thesolution of the FBSDE is affected by the terminal condition119884119879

= 120585(119883119879) As is well known there exist a number

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 365240 11 pageshttpdxdoiorg1011552014365240

2 Mathematical Problems in Engineering

of parametric and nonparametric methods to deal withestimation and test for the OSDE However these methodscannot be directly employed to infer the BSDE and FBSDEbecause the two models are related to a terminal conditionForward-backward stochastic differential equations are usedin biology systems mathematical finance insurance realestate multiagent and network control See Antonelli [3]Wang et al [4] Zhang and Li [5] and so on

For the FBSDE defined above the statistical inference wasinvestigated initially by Su and Lin [6] and Chen and Lin [7]Furthermore by financial and ecological problems a relevantstatistical model was proposed by Lin et al [8] Howeverthey did not take the terminal condition into account inthe inference procedure In the framework of the FBSDEmentioned above the terminal condition is additional whichis not nested into the equation Thus there is an essentialdifficulty to use the terminal condition to refine the inferenceprocedure

As a result their methods fail to cover the full problemsgiven in the FBSDE Zhang and Lin [9] proposed twoterminal-dependent estimationmethods via terminal controlvariable for the integral form models of FBSDE Howeverthey only considered the parametric form of the generator 119892in their paper

This paper intends to explore the method to fulfill theterminal-dependent inference quasi-instrumental variablemethods It is worth mentioning that the key point of ourmethod is the use of the terminal condition informationrather than neglecting it This change leads to a completelynew work among the existing researches The key techniquein ourmethod is the use of quasi-instrumental variable whichis similar but not the same as instrumental variable (IV) It isknown that IV is widely employed in applied econometrics toachieve identification and carry out estimation and inferencein the model containing endogenous explanatory variablesor panel data see Hsiao [10] for an overview of the relevantstatistical inference and econometric interpretation and seeHall and Horowitz [11] for recent work on nonparametricinstrumental variable estimation

Through the backward equation (2) of FBSDE we get aregression model To use the terminal condition informa-tion we put the terminal condition as a quasi-instrumentalvariable and introduce it into our model However when aconstraint is appended artificially the original model maychange to be biased in the sense of 119864(119885

119904119889119861

119904| 119883

119904 120585) = 0

because the constraint condition influences the increase trendof wealth so that 119885

119904119889119861

119904may deviate from the original center

zero in other words due to the constraint the trajectoryof 119884

119904may departure from the original expectation so that

119885119904119889119861

119904cannot be regarded as errorTherefore some problems

arise naturally including how to correct the bias of the modeland how to construct the constraint-dependent estimationTo solve these problems we will use remodeling method todraw terminal condition into differential equation similarbut not the same as IV called quasi-instrumental variablemethods in other words the terminal condition 120585 enters intothe equation as a control variable This remodeling methodtakes advantage of the terminal information naturally and theestimator performs quite well

We use the nonparametric form of the generator 119892 inmodel (2) because the correct FBSDEs model for a specifictopic can neither be provided automatically by financialmarket nor be derived from theory of mathematical financeand in lack of prior information about the structure ofa model nonparametric inference can provide a flexibleas well as robust description of a data-generating processEven in some cases when parametric models are availablenonparametricmethods are still employed to avoid themodelmisspecification that may lead to large errors in optionpricing and other problems from financial market So weadopt the nonparametric form that can endow the model (2)with flexibility and robustness

Note that 119885119904is usually unobservable and 119892 cannot be

completely specified in the financial marketThe problems ofinterest are therefore to give both proper estimations of thegenerator 119892 and the process 119885

119904based on the observed data

(119883119904 119884

119904) and the terminal expectation 120585

The remainder of the paper is organized as follows InSection 2 the FBSDE is rebuilt as a nonparametricmodel thatcontains the terminal condition as a quasi-instrumental vari-able Consequently a terminal-dependent estimation proce-dure is proposed Next we discuss the asymptotic propertiesof the newly proposed estimations in Section 3 Simulationstudy is proposed in Section 4 to illustrate our methods Theproofs of the theorems are presented in Appendix

2 Model and Method

In this section we propose a nonparametric estimator withthe help of quasi-instrumental variable

21 Model and Its Statistical Version We begin the followingoriginal model by combining (2)-(3)

119889119884119904= minus119892 (119904 119883

119904 119884

119904 119885

119904) 119889119904 + 119885

119904119889119861

119904 119884

119879= 120585

119889119883119904= 120583 (119904 119883

119904) 119889119905 + 120590 (119904 119883

119904) 119889119861

119904 119904 isin [119905 119879]

(4)

where 119861119905is the standard Brownian motion and 120585 is a R-

valued Borel function Here the generator 119892 is a function of119904 119883

119904 119884

119904 and 119885

119904 For the FBSDEs model (4) only one of

the backward components 119884119904 and the forward components

119883119904 can be observed Another backward component 119885

119904is

totally unobservable Furthermore the adapted process 119885119904

and terminal condition could be indicated as a function of119883119904In this section we present the statistical structure of

FBSDEs by taking advantage of quasi-instrumental variableand obtain the consistent asymptotically normal estimatorsof 119892 and 119885

119904based on observed data 119883

119904 119884

119904 and the terminal

condition 120585

22 Remodeling for Model (4) To construct terminal-dependent estimation for the generator 119892 and process 119885

119904

the key technique is how to plug the terminal condition intothe equation When 120585 is plugged into the model we call itthe quasi-IV similar but not the same as IV Evidently theproperty of Brownian motion shows that 119864(119885

119904119889119861

119904| 119883

119904) = 0

Mathematical Problems in Engineering 3

but 119864(119885119904119889119861

119904| 119883

119904 120585) = 0 which means drawing the terminal

control directly into the equation as the condition should notbe encouraged at the cost of model bias Rewriting the firstequation of (4) enables us to construct an unbiased model

119889119884119904= minus119892 (119904 119883

119904 119884

119904 119885

119904) 119889119904 + 119898 (119883

119904 120585) + 119880

119904 (5)

where 119898(119883119904 120585) = 119864(119885

119904119889119861

119904| 119883

119904 120585) 119880

119904= 119885

119904119889119861

119904minus

119898(119883119904 120585) and 119864(119880

119904| 119883

119904 120585) = 0 The newly defined

model (5) together with the second equation in (4) can bethought of as a quasi-IV FBSDE Because the equation in(5) contains the terminal condition 120585 we can construct theterminal-dependent estimation From the above definitionswe see that by bias correction the original model changesto be an additive nonparametric model with nonparametriccomponents minus119892(119904 119883

119904 119884

119904 119885

119904)119889119904 and 119898(119883

119904 120585) It shows that

when terminal condition is regarded as a quasi-IV and thenappended to the model the result model is unbiased andchanges to be nonparametric additive model

23 Estimation for 119885119904 Before estimating the model function

119898(119909119904 120585) and the generator 119892 we need to estimate 119885

119904firstly

because 119885119904is unobservable and it will be seen that the

estimators of the model function 119898(119909119904 120585) and the generator

119892 depend on 119885119904 Since the distribution of 120585 is supposed to

be known let 120585119894 1 le 119894 le 119896 for 119896 ge 1Δ be a sample of

120585 Suppose that for each terminal data 120585119895and equally spaced

time points 119904119894= 119904

1+(119894minus1)Δ 119894 = 1 119899 sube [0 119879] we record

the observed time series data119883

119904119894 119895 119884

119904119894 119895 119894 = 1 119899 119895 = 1 119896

= 119883119894119895 119884

119894119895 119894 = 1 119899 119895 = 1 119896

(6)

At any time point 119904 isin [119905 119879] 119885119905119909

119904 denoting 119885

119904and satisfying

the initial condition (119905 119909) is a determined function of 119883119905119909

119904

As was shown by Su and Lin [6] and Chen and Lin [7] wecan adopt a difference-based method to approximate 1198852 as

(119885119905119883119905

119904)2

=1

Δ119864(119884

119905+Δ119883119905+Δ

119904+Δminus 119884

119905119883119905

119904| 119883

119905 119905)

2

+ 119874 (Δ) (7)

It shows that the numerical approximation error to 1198852

119905

converges to zero at rate of order 119874119901(Δ)

For each 120585119895 if119885

119905depends on 119905 only via variable119883

119905 by (7)

and N-W kernel estimation method we estimate 1198852

119905at 119909

0by

1198852

1199090 119895=

sum119899minus1

119894=1Δminus1

(119884119894+1119895

minus 119884119894119895)2

119870ℎ119883

(119883119894119895

minus 1199090)

sum119899minus1

119894=1119870ℎ119883

(119883119894119895

minus 1199090)

(8)

Otherwise we estimate 1198852

119905at (119909

0 1199050) by

1198852

1199090 1199050 119895

=sum119899minus1

119894=1Δminus1

(119884119894+1119895

minus 119884119894119895)2

119870ℎ119883

(119883119894119895

minus 1199090)119870

ℎ119905(119905119894minus 119905

0)

sum119899minus1

119894=1119870ℎ119883

(119883119894119895

minus 1199090)119870

ℎ119905(119905119894minus 119905

0)

(9)

where 119870ℎ119883

= 119870(sdotℎ119883)ℎ

119883and 119870

ℎ119905= 119870(sdotℎ

119905)ℎ

119905 119870(sdot) are reg-

ular kernel functions with ℎ119883and ℎ

119905being the corresponding

bandwidths

24 Estimation for 119898(119883119904120585) After plugging the estimator 119885

119904

into model (5) we still need to consider inference of119898(119909119904 120585)

As we all know the nonparametric function 119898(119883119904 120585) in (5)

can be acquired as 119898(119883119904 120585) = 119864(119889119884

119904+ 119892(119904 119883

119904 119884

119904 119885

119904)119889119904 |

119883119904 120585) We note that 119892(119904 119883

119904 119884

119904 119885

119904)119889119904 is a higher order

infinitesimal of 119885119904119889119861

119904when Δ tends to zero Under this

situation if 119892(119904 119883119904 119884

119904 119885

119904)119889119904 is ignored then

119898(119883119904 120585) ≐ 119864 (119889119884

119904| 119883

119904 120585) (10)

It implies that we can use ordinary nonparametric methodto estimate function 119898 For example we use the N-Wordinary nonparametric method to estimate119898(119883

119904 120585) valued

at (1199090 120585

0)

(1199090 120585

0)

=

sum119899minus1

119894=1sum119898

119895=1(119884

119894+1119895minus 119884

119894119895)119870

ℎ119883(119883

119894119895minus 119909

0)119870

ℎ120585(120585

119894119895minus 120585

0)

sum119899minus1

119894=1sum119898

119895=1119870ℎ119883

(119883119894119895

minus 1199090)119870

ℎ120585(120585

119894119895minus 120585

0)

(11)

where 119870ℎ119883

= 119870(sdotℎ119883)ℎ

119883and 119870

ℎ120585= 119870(sdotℎ

120585)ℎ

120585are regular

kernel functions with ℎ119883and ℎ

120585being the corresponding

bandwidths

25 Estimation for Generator 119892 As was shown in the non-parametric instrumental variables estimator of Hall andHorowitz [11] (hereinafter HH) we can adopt a nonpara-metric quasi-instrumental variables estimation to estimatethe generator 119892 So in the section we summarize the HHestimator of 119892 in the model

119864 [minus119889119884119905minus 119898 (119883

119905 120585) | 119883

119905 120585] = 119864 [119892 (119905 119883

119905 119884

119905 119885

119905) 119889119905 | 119883

119905 120585]

(12)

Since (1199090 120585

0) and 119885

2

1199090 119895are the consistent estimator of

119898(1199090 120585

0) and 119885

2

1199090 119895 respectively we use them instead of

119898(119883119904 120585) and 119885

119904in the above model and we get

119864 [minus119889119884119904minus (119883

119904 120585) | 119883

119904 120585]

= 119864 [119892 (119904 119883119904 119884

119904 119885

119904) 119889119905 | 119883

119904 120585]

(13)

Because 119885119904is function of 119883

119904and 119884

119904 for simplicity of

presentation we denote 119892(119904 119883119904 119884

119904 119885

119904) = 119892(119883

119904 119884

119904)Thus the

model becomes

119864 [minus119889119884119904minus (119883

119904 120585) | 119883

119904 120585] = 119864 [119892 (119883

119904 119884

119904) 119889119905 | 119883

119904 120585]

(14)

Let Y119894= ((119884

119894+Δminus 119884

119894) minus (119883

119894 120585))Δ X

119894= 119883

119894 Z

119894= 119884

119894

W = 120585 and U119894= 119881

119894radicΔ the model becomes

Y119894= 119892 (X

119894Z

119894) + U

119894 119864 (U

119894| X

119894W

119894) = 0 (15)

It is assumed that the support of (XZW) is containedin [0 1]

3 This assumption can always be satisfied by ifnecessary carrying outmonotone increasing transformationsofXZ andW For example one can replaceXZ andW by

4 Mathematical Problems in Engineering

Φ(X) Φ(Z) and Φ(W) where Φ is the normal distributionfunction We take (Y XZWU) to be a vector where Y

and U are scalars X and W are supported on [0 1] and Z

is supported on [0 1] The model is

Y119894= 119892 (X

119894Z

119894) + U

119894 119864 (U

119894| Z

119894W

119894) = 0 (16)

where (Y119894X

119894Z

119894W

119894U

119894) for 119894 ge 1 are independent and

identically distributed as (Y XZWU) Thus X and Z areendogenous and exogenous explanatory variables respec-tively Data (Y

119894X

119894Z

119894W

119894U

119894) for 1 le 119894 le 119899 are observed

Let119891XZW denote the density of (XZW) write119891Z for thedensity of Z and for each 119909

1 119909

2isin [0 1]

119901 and put

119905119911(119909

1 119909

2) = int119891XZW (119909

1 119911 119908) 119891XZW (119909

2 119911 119908) 119889119908 (17)

Define the operator 119879119911on 119871

2[0 1]

119901 by

(119879119911120595) (119909) = int 119905

119911(120585 119909) 120595 (120585) 119889120585 (18)

It may be proved that for each 119911 for which 119879minus1

119911exists

119892 (119909 119911)

= 119891Z (119911) 119864W|Z

times 119864 (Y | Z = 119911W) (119879minus1

119911119891XZW) (119909 119911W) | Z = 119911

(19)

where 119864W|Z denotes the expectation with respect to thedistribution of W conditional on Z In this formulation(119879

minus1

119911119891XZW)(119909 119911W) denoted the result of applying 119879minus1

119911to the

function 119891XZW(sdot 119911W) and evaluating the resulting functionat 119909

To construct an estimator of 119892(119909 119911) given ℎ gt 0 and 119909 =

119909(1) and 120585 = 120585

(1) let 119870ℎ(119909 120585) = 119870

ℎ(119909

(119895)

120585(119895)

) put 119870ℎ(119911 120585)

analogously for 119911 and 120585 let ℎ119909 ℎ

119911gt 0 and define

119891XZW (119909 119911 119908)

=1

119899ℎ2119909ℎ119911

119899

sum

119894=1

119870ℎ119909

(119909 minusX119894 119909)119870

ℎ119911(119911 minus Z

119894 119911)119870

ℎ119909(119908 minusW

119894 119908)

119891minus119894

XZW (119909 119911 119908)

=1

(119899 minus 1) ℎ2119909ℎ119911

sum

1le119895le119899119895 = 119894

119870ℎ119909

(119909 minusX119895 119909)

times 119870ℎ119911

(119911 minus Z119895 119911)119870

ℎ119911(119908 minusW

119895 119908)

119911(119909

1 119909

2) = int119891XZW (119909

1 119911 119908) 119891XZW (119909

2 119911 119908) 119889119908

(119911120595) (119909 119911 119908) = int

119911(120585 119909) 120595 (120585 119911 119908) 119889120585

(20)

where120595 is a function from1198773 to a real lineThen the estimator

of 119892(119909 119911) is

119892 (119909 119911) =1

119899

119899

sum

119894=1

(+

119911119891minus119894

XZW) (119909 119911W119894) 119884

119894119870ℎ119911

(119911 minus Z119894 119911) (21)

3 Asymptotic Results

In this section we study the asymptotic properties of ourproposed estimators All proofs are presented in Appendix

31 Asymptotic results of119885119904 To give the asymptotic results of

119885119904 we need the following conditions

(a) 1198831 119883

119899are 120588-mixing dependent namely the 120588-

mixing coefficients 120588(119897) satisfy 120588(119897) rarr 0 as 119897 rarr infinwhere

120588 (119897) = sup119864(119883119894+119897119883119894)minus119864(119883119894+119897)119864(119883119894) = 0

1003816100381610038161003816119864 (119883119894+119897119883119894) minus 119864 (119883

119894+119897) 119864 (119883

119894)1003816100381610038161003816

radicVar (119883119894+119897)Var (119883

119894)

(22)

with119883119894= 119883(119905

119894)

(b) |119885119894| le 119862 (a s) uniformly for 119894 = 1 119899 where 119862 is a

positive constant and 119885119894= 119885(119905

119894)

(c) The continuous kernel function 119870(sdot) is symmetricabout 0 with a support of interval [minus1 1] and

int

1

minus1

119870 (119906) 119889119906 = 1 1205902

119870= int

1

minus1

1199062

119870 (119906) 119889119906 = 0

int

1

minus1

|119906|119895

119870119896

(119906) 119889119906 lt infin for 119895 le 119896 = 1 2

(23)

Condition (a) is commonly used for weakly dependentprocess see for example Kolmogorov and Rozanov [12]Bradley and Bryc [13] Lu and Lin [14] and Su and Lin [6]Condition (b) is also reasonable because as is shown by (10)119885119905can be regarded as the deviation between the adjacent two

observations Condition (c) is standard for nonparametrickernel function

Theorem 1 Besides conditions (a) (b) and (c) let119883

1 119883

119899 be an observation sequence on a stationary

120588-mixing Markov process with the 120588-mixing coefficientssatisfying 120588(119897) = 120588

119897 for 0 lt 120588 lt 1 Furthermore 1198831 119883

119899

have a common and probability density 119901(119909) and for eachinterior point 119909

0in the support of 119901(sdot) 119901(119909

0) gt 0 1198852

(1199090) gt 0

the functions 119901(119909) and 119885(119909) have continuous two derivativesin neighborhood of 119909

0 As 119899 rarr infin such that 119899ℎ rarr infin

119899ℎ5

rarr 0 and 119899ℎΔ2

rarr 0 then

radic119899ℎ (1198852

(1199090) minus 119885

2

(1199090))

119889

997888rarr (01198854

(1199090) 119869

119870

119901 (1199090)

) (24)

where 119869119870= int

1

minus1

1198702

(119906)119889119906 lt infin

The asymptotic result in Theorem 1 is standard fornonparametric kernel estimator and here undersmoothing isused to eliminate asymptotic bias

32 Asymptotic results of 119892(119909119911) This section gives con-ditions under which the HH estimator of the generator

Mathematical Problems in Engineering 5

119892 is asymptotically distributed as 119873(0 119868) The followingadditional notations are used

Define U119894

= Y119894

minus 119892(X119894Z

119894) 119878

1198991(119909 119911) =

119899minus1

sum119899

119894=1U119894+

119891(minus119894)

XZW(119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911) and 119878

1198992(119909 119911) =

119899minus1

sum119899

119894=1119892(X

119894Z

119894)

+

119891(minus119894)

XZW(119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911) Then

119892(119909 119911) = 1198781198991(119909 119911) + 119878

1198992(119909 119911) Define 119879+

= (119879+ 119886119899119868)

minus1 Write

1198781198991

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894(119879

+

119891XZW) (119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911)

+ 119899minus1

119899

sum

119894=1

U119894(

+

119891(minus119894)

XZW minus 119879+

119891XZW)

times (119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911)

= 11987811989911

(119909 119911) + 11987811989912

(119909 119911)

(25)

Define 119881119899(119909 119911) = 119899

minus1 Var[U(119879+

119891XZW)(119909 119911W)] It followsfrom a triangular array version of the Lindeberg-Levy centrallimit theorem that 119878

11989911(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0 1) as 119899 rarr

infin Therefore [119892(119909 119911) minus 119892(119909 119911)]radic119881119899(119909 119911)rarr

119889

119873(0 1) if[119878

11989912(119909 119911) + 119878

1198992(119909 119911) minus 119892(119909 119911)]radic119881

119899(119909 119911) = 119900

119901(1)

Assumption 2 The data Y119894X

119894 Z

119894W

119894are independently and

identically distributed as (Y XXW) where (XZW) issupported on [0 1]

3 and 119864[Y minus 119892(XZ) | WZ] = 0

Assumption 3 The distribution of (XZW) has a density119891XZW with respect to Lebesgue measure Moreover 119891XZW is119903 times differentiable with respect to any combination of itsarguments where derivatives at the boundary of [0 1]3 aredefined as one sided derivativesThe derivatives are boundedin absolute value by 119862 In addition 119892 is 119903 times differentiableon [0 1]

2 with derivatives at 0 and 1 defined as one sidedThe derivatives of 119892 are bounded in absolute value by 119862 Inaddition 119864[Y 2

| XZW] le 119862 and 119864[Y 2

| XZW] le 119862 and119864[U2

| ZW] ge 119862119880for some finite constant 119862

119880

Assumption 4 The constants 120572 and 120573 satisfy 120572 gt 1 120573 gt 12and 120573 minus 12 le 120572 lt 2120573 Moreover 119887

119895le 119862119895

minus120573 119895minus120572 le 119862120582119895 and

suminfin

119896=1|119889

119911119895119896| le 119862119895

minus1205722 for all 119895 ge 1 In addition there are finitestrictly positive constants 119862

1205821and 119862

1205822 such that 119862

1205821le 120582

119895le

1198621205822119895minus120572 for all 119895 ge 1

Assumption 5 The tuning parameters 119886119899and ℎ satisfy 119886

119899≍

119899minus(120588120572)(2120573+120572) and ℎ ≍ 119899

minus1 where 119903 isin [1198601015840

2 119860

1015840

3]

Assumption 6 119870ℎdenotes a generalized kernel function with

the properties 119870ℎ(119906 119905) = 0 if 119906 gt 119905 or 119906 lt 119905 minus 1 for all

119905 isin [0 1]ℎminus(119895+1)

int119905minus1

119905119906119895

119870ℎ(119906 119905)119889119906 = 1 if 119895 = 0 else 0 if

1 le 119895 le 119903 minus 1 For each 120585 isin [0 1] 119870ℎ(ℎ 120585) is supported

on [(120585 minus 1)ℎ 120585ℎ] cap 120581 where 120581 is a compact interval notdepending on 120585 Moreover

supℎgt0120585isin[01]119906isin120581

119870ℎ(ℎ119906 120585) |lt infin (26)

Assumption 7 Consider 119864W[119879+

119891XZW(119909 119911W)]2

≍

119864W[119879+

119891XZW(sdot sdotW)]2 and 119864W[119879

+

119891XZW(sdot sdotW)]2

≍

int1

0

119879+

119891XZW(sdot sdotW)2

119889119908

Theorem 8 Let Assumptions 2ndash7 hold Then

119892 (119909 119911) minus 119892 (119909 119911)

radic119881119899(119909 119911)

997888rarr119889

119873(0 119868) (27)

holds except possibly on a set of (119909 119911) values whose Lebesgueis 0

Corollary 9 Let Assumptions 2ndash7 hold And if 119881119899(119909 119911) is

replaced with the consistent estimator

119899(119909 119911) = 119899

minus1

119899

sum

119894=1

U2

119894[

+

119891minus119894

119909119908(119911W

119894)119870

119902ℎ119911(119911 minus Z

119894 119911)]

2

(28)

where U119894= Y

119894minus 119892(X

119894Z

119894) This yields the Studentized statistic

[119892(119909 119911) minus 119892(119909 119911)]radic119899(119909 119911) Then

119892 (119909 119911) minus 119892 (119909 119911)

radic119899(119909 119911)

997888rarr119889

119873(0 119868) (29)

holds except possibly on a set of (119909 119911) values whose Lebesgueis 0

As was shown in the remark given in the previoussection even the conditional mean of error of the model isnonzero and the newly proposed estimation is consistentbecause of themixing dependency for details see the proof ofTheorem 8 Furthermore because of the terminal conditionthe asymptotic variance is larger than that without the use ofthe terminal condition

4 Simulation Studies

In this section we investigate the finite-sample behaviors bysimulation

Example 10 We consider a simple FBSDE as

119889119884119905= (

120583 minus 119903

120590119885119905+ 119903119884

119905)119889119905 + 119885

119905119889119861

119905

≜ (119887119884119905+ 119888119885

119905) + 119885

119905119889119861

119905 119884

119879= 120585

(30)

where119883119905is Geometric Brownian motion for modeling stock

price satisfying

119889119883119905= 120583119883

119905119889119905 + 120590119883

119905119889119861

119905 119883

0= 119909 (31)

while the riskless asset is the same as formula (31) 119889119875119905

=

1199031198750119889119905

Firstly let 120583 = 01 120590 = 001 Δ = 012 119899 = 300119879 = 366 and 119899

0= 119899

1= 10 Obviously119885

119905= 119899

1120590119883

119905We adopt

Epanechnikov kernel defined by119870(119906) = 34(1minus1199062

)119868(|119906| le 1)

6 Mathematical Problems in Engineering

012

01

008

006

004

002

0

Curve of ZEstimated curve of Z

0 5 10 15 20 25 30 35 40

(a)

Curve of gEstimated curve of g

0 5 10 15 20 25 30 35 40

14

12

1

08

06

04

02

0

(b)

Figure 1 The real lines are the true curves of 119885119905and function 119892(119905) respectively and the dashed ones are estimated curves for them in

Example 10

where 119868(sdot) is the indicator function For bandwidth selectionvarious data-driven techniques have been developed suchas cross-validation the plug-in method and the empiricalbias method However these useful tools require additionalcomputation intensiveness In our simulation we simplyapply the rule of thumb bandwidth selector For bandwidthselection bandwidth ℎ = std(119909)119899minus15 The values of thetuning parameters are 119886

119899= 005 120572 = 12 120573 = 1 Figure 1

presents the estimated curves for diffusion 119885119905and drift 119892 by

one simulation

Example 11 According to the theory ofmathematical financewe represent a European call option by the following FBSDEsmodel

119889119883119904= 119887119883

119904119889119904 + 120590119883

119904119889119882

119904

119889119884119904= [119903119884

119904+ (119887 minus 119903) 120590

minus1

119885119904] 119889119904 + 119885

119904119889119882

119904

1198830= 119909 119884

119879= (119883

119879minus 119870)

+

119904 isin [0 119879]

(32)

Here 1198831199040le119904le119879

and 1198841199040le119904le119879

are the price processes of thestock and the option respectively and119870 is the striking priceat the expiration date 119879 119883

1199040le119904le119879

follows the geometricBrownian motion as

119889119883119904= 119887119883

119904119889119904 + 120590119883

119904119889119882

119904

1198830= 119909 119904 isin [0 119879]

(33)

We use the Euler scheme to generate the price series ofthe stock as

119883119894+1

minus 119883119894= 119887119883

119894Δ + 120590119883

119894Δ12

120598119894 119894 = 0 119899 minus 1 (34)

where 120598119894119899minus1

119894=0is an iid series with standard normality

The price series by Black Scholes formula is part of thesolution of the FBSDEs above at discrete time points that is

119884119894= 119883

119894119873(119889

119894

+) minus 119890

minus119903(119899minus119894)Δ

119870119873(119889119894

minus) (35)

which together with

119885119894= 120590119883

119894119873(119889

119894

+) (36)

gives us data generating formulae where

119873(119910) =1

radic2120587int

119910

minusinfin

119890minus11990922

119889119909 (37)

is a cumulative normal function and

119889119894

plusmn=ln (119883

119894119870) + (119903 plusmn 120590

2

2) ((119899 minus 119894) Δ)

120590radic(119899 minus 119894) Δ (38)

We produce the true curve of the drift coefficient by

119892119894= minus119903119884

119894minus (119887 minus 119903) 120590

minus1

119885119894 (39)

We first apply formulas (21) and (11) to estimate 119892119894and

1198852

119894 respectively We adopt Epanechnikov kernel defined by

119870(119906) = 34(1 minus 1199062

)119868(|119906| le 1) where 119868(sdot) is the indicatorfunction For bandwidth selection we simply apply the ruleof thumb bandwidth selector

ℎ = constant times std (1198840 119884

119899minus1) 119899

minus15 (40)

to implement the estimationLet 119870 = 110 119883

0= 100 119887 = 01 120590 = 018 119903 = 008

119879 = 60 and Δ = 1100 The bandwidth parameters ℎ = 606

and ℎ = 067 are used for estimation of119892119904and119885

119904 respectively

The values of the tuning parameters are 119886119899= 005 120572 = 12

and 120573 = 1 To see the performance of our estimationmethodthe simulated and the estimated curves of the two coefficientsof the backward equation are displayed in Figures 2 and 3

Mathematical Problems in Engineering 7

0

0

10 20 30 40 50 60minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

minus100

Curve of gEstimated curve of g

Figure 2 The simulated curve and the estimated curves of 119892119904in

Example 11

00

10 20 30 40 50 60

Curve of ZEstimated curve of Z

800

700

600

500

400

300

200

100

Figure 3 The simulated curve and the estimated curves of 119885119904in

Example 11

Appendix

A Proofs

Proof of Theorem 1 Denote C = 1198831 119883

119899 By the

Taylor expansion and formula (8) we have

119864 (1198852

(1199090) | C)

=sum119899minus1

119894=1Δminus1

119870ℎ(119883

119894minus 119909

0) 119864 ((119884

119894+1minus 119884

119894)2

| C)

sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0)

=sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0) (119885

2

119894+ 119874 (Δ))

sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0)

=int119870

ℎ(119883

119894minus 119909

0) (119885

2

(119909)+119874 (Δ))119901 (119909) 119889119909 (1+ 119874119901(119899ℎ)

minus12

)

int119870ℎ(119883

119894minus 119909

0) 119901 (119909) 119889119909 (1+119874

119901(119899ℎ)

minus12

)

= ( (1198852

(1199090) + 119874 (Δ))

times (119901 (1199090) + (12) ℎ

2

119901(2)

(1199090) 120590

2

119870+ 119900 (ℎ

2

))

times (1 + 119874119901(119899ℎ)

minus12

) )

times ( (119901 (1199090) + (12) ℎ

2

119901(2)

(1199090) 120590

2

119870+ 119900 (ℎ

2

))

times (1 + 119874119901(119899ℎ)

minus12

) )

minus1

= 1198852

(1199090) +

119901(2)

(1199090)

2119901 (1199090)ℎ2

1198852

(1199090) 120590

2

119870+ 119900 (ℎ

2

) + 119874 (Δ)

(A1)

Furthermore

Var (1198852

(1199090) | C)

=1

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

times

119899minus1

sum

119894=1

Δminus2

1198702

ℎ(119883

119894minus 119909

0)Var ((119884

119894+1minus 119884

119894)2

| C)

+

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

Δminus2 cov (119870

ℎ(119883

119894minus 119909

0) (119884

119894+1minus 119884

119894)

119870ℎ(119883

119894+119896minus 119909

0) (119884

119894+119896+1minus 119884

119894+119896) | C)

(A2)

From the conditions of Markov process and 120588-mixing coeffi-cient1003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

Δminus2 cov ( 119870

ℎ(119883

119894minus 119909

0) (119884

119894+1minus 119884

119894)

119870ℎ(119883

119894+119896minus 119909

0) (119884

119894+119896+1minus 119884

119894+119896) )

1003816100381610038161003816100381610038161003816100381610038161003816

=1

(119899 minus 1)2

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

10038161003816100381610038161003816100381610038161003816119864 ((Δ)

minus2

(119884119894+1

minus 119884119894)2

(119884119894+119896+1

minus 119884119894+119896

)2

times (119870ℎ(119883

119894minus 119909

0) minus 119864 (119870

ℎ(119883

119894minus 119909

0)))

times(119870ℎ(119883

119894+119896minus119909

0)minus119864 (119870

ℎ(119883

119894+119896minus119909

0))))

10038161003816100381610038161003816100381610038161003816

=1

(119899 minus 1)2

10038161003816100381610038161003816100381610038161003816119864 (119885

2

1198941198852

119894+119897(119870

ℎ(119883

119894minus 119909

0) minus 119864 (119870

ℎ(119883

119894minus 119909

0)))

times (119870ℎ(119883

119894+119896minus 119909

0) minus 119864 (119870

ℎ(119883

119894+119896minus 119909

0))))

10038161003816100381610038161003816100381610038161003816

+ 119874 (Δ)

le119862

(119899 minus 1)2

ℎ

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

120588119896

= 119874(1

119899ℎ) = 119900 (1)

(A3)

8 Mathematical Problems in Engineering

Note that (119884119894+1

minus 119884119894)radicΔ = 119892(119905

119894 119884

119894 119885

119894)radicΔ + 119885

119894120578119894 where

119864(120578119894) = 0 Var(120578

119894) = 1 Thus Var((119884

119894+1minus119884

119894)radicΔ) = 119885

4

119894+119874(Δ)

and furthermore

Var (1198852

(1199090) | C)

=sum119899minus1

119894=1Δminus2

1198702

ℎ(119883

119894minus 119909

0)Var ((119884

119894+1minus 119884

119894)2

| C)

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

+ 119874119901(1)

=sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0) (119885

4

(1199090) + 119874 (radicΔ))

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

+ 119874119901(1)

=1198854

(1199090) 119869

119870+ 119874 (radicΔ)

119899ℎ119901 (1199090)

(1 + 119874119901(119899ℎ)

minus12

)

(A4)

To our interest both the conditional expectation and varianceare independent onC so the condition could be erased

From Lemma A1 of Politis and Romano [15] and therelation between the 120572-mixing condition and the 120588-mixingcondition (eg Theorem 111 of Lu and Lin [14]) we canensure that (119884

119894+1minus 119884

119894)2

119894 = 1 119899 minus 1 is a 120588-mixingdependent process and the mixing coefficient denoted by120588119884(119897) satisfies

infin

sum

119896=1

120588119884(2

119896

) le 119862

infin

sum

119896=1

120588 (2119896

) =

infin

sum

119896=1

1205882119896

lt infin (A5)

where119862 is a positive constant Finally we use the central limittheorems for 120588-mixing dependent process (eg Theorem401 of Lu and Lin [14]) to complete this proof

Proof of Theorem 8 Theorem 8 follows from proving that1198781198991(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0 1198682) and [119878

1198992(119909 119911) minus 119892(119909 119911)]

radic119881119899(119909 119911) = 119900

119901(1) except possibly if (119909 119911) belongs to a

set of Lebesgue measure 0 The first result is established inLemma A1 and the second is established in Lemma A2Throughout this Appendix ldquofor almost every (119909 119911)rdquo meansldquofor every (119909 119911) isin [0 1]

2 except possibly a set of Lebesguemeasure 0rdquo We make repeated use of the fact that if 1198641205952 =

119874(119899minus119904

) for some 119904 gt 0 then120595(119909 119911) = 119900119901(119899

minus119904

) for almost every(119909 119911)

Lemma A1 (asymptotic normality of 1198781198991(119909 119911)radic119881

119899(119909 119911))

Let Assumptions 2ndash7 hold Then 1198781198991(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0

1198682) for almost every (119909 119911)

Proof Define 11987811989911

(119909 119911) = 119899minus1

sum119899

119894=1U119894(119879

+

119891XZW)(119909 119911W119894)

1198601198992

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[119879

+

(119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

1198601198993

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[(

+

minus 119879+

) 119891XZW] (119909 119911W119894)

1198601198994

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[(

+

minus 119879+

) (119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

(A6)

Then 1198781198991(119909 119911) = 119878

11989911(119909 119911)+119860

1198992(119909 119911)+119860

1198993(119909 119911)+119860

1198994(119909 119911)

11987811989911

(119909 119911)radic119881119899(119909 119911)rarr

119889

119873(0 1198682) by a triangular array version

of the Lindeberg-Levy central limit theorem The proof ofthe triangular-array version of the theorem is identical to theproof of the ordinary Lindeberg-Levy theorem The lemmafollows if we can prove that 119860

119899119895(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for

119895 = 2 3 4 and almost every (119909 119911) isin [0 1]2

Assumption 7 and arguments like those leading to (62)of HH [11] show that

∬

1

0

119881119899(119909 119911) 119889119909 119889119911 ≍ 119899

minus[2120573+120572minus120588(120572+1)](2120573+120572)

(A7)

It follows from the Cauchy-Schwartz inequality 119864(119891(minus119894)

XZW minus

119891XZW) = 119874(ℎ1015840

) and Var(119891(minus119894)

XZW) = 119874[1(119899ℎ2

)] that

11986410038171003817100381710038171198601198992

10038171003817100381710038172

= 119874(1

1198992ℎ21198862119899

+ℎ2119903

1198991198862119899

) (A8)

Therefore it follows from Assumptions 5 and 7 that119860

1198992(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909 119911) Now

consider 1198601198993(119909 119911) Define the operator Δ = minus 119879 Then

1198601198993

(119909 119911) = minus ( + 119886119899119868) Δ119860

1198991(119909 119911) (A9)

Therefore the Cauchy-Schwartz inequality gives

11986410038171003817100381710038171198601198992

10038171003817100381710038172

le 11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

119864100381710038171003817100381711987811989911

10038171003817100381710038172

= 11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

∬

1

0

119881119899(119909 119911) 119889119909 119889119911

(A10)

HH show that

11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

= 119874(1

119899ℎ1198862119899

+ℎ2119903

1198862119899

) (A11)

Therefore it follows from Assumptions 5 and 7 that119860

1198993(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909 119911) Finally

some algebra shows that

1198601198994

(119909 119911) = minus( + 119886119899119868)

minus1

Δ1198601198992

(119909 119911) (A12)

Therefore 1198601198994(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909

119911) follows from (A11) and 1198601198992(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1)

Lemma A2 (asymptotic negligibility of 1198781198992(119909 119911) minus 119892(119909 119911))

Let Assumptions 2ndash7 hold Then 1198781198992(119909 119911) minus 119892(119909 119911)

radic119881119899(119909 119911) = 119900

119901(1) for almost every (119909 119911)

Mathematical Problems in Engineering 9

Proof Define

119863119899(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times (119891XZW minus 119891XZW) (119909 119911 119908) 119889120579 119889120578 119889119908

1198601198991

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894 119885

119894) (119879

+

119891XZW) (119909 119911W119894)

(A13)

Redefine

1198601198992

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

minus 119863119899(119909 119911)

1198601198993

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) 119891XZW] (119909 119911W119894) + 119863

119899(119909 119911)

1198601198994

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) (119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894)

(A14)

Then 1198781198992(119909 119911) = sum

4

119895=1119860

119899119895(119909 119911) Arguments identical to

those used to derive (62) and (63) of HH [11] show that119864119860

1198991minus 119892

2

= 119874[119899minus120588(21205731)(2120573+120572)] and

∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874119899minus[2120573+120572minus120588(120572+1)](2120573+120572)

(A15)

Therefore it follows from Assumptions 5 and 7 that

[1198641198601198991

(119909 119911) minus 119892 (119909 119911)]

radic119881119899(119909 119911)

= 119900 (1) (A16)

119881minus1

119899(119909 119911)∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874 (1) (A17)

for almost every (119909 119911)Now consider 119860

1198992(119909 119911) Define

119863119899119894(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times(119891(minus119894)

XZWminus119891XZW)(119909 119911 119908) 119889120579 119889120578 119889119908

11986011989921

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894) minus 119863

119899119894(119909 119911)

(A18)

and 11986011989922

(119909 119911) = 119899minus1

sum119899

119894=1[119863

119899119894(119909 119911) minus 119863

119899(119909 119911)] HH show

that 11986411986011989921

2

= 119874((ℎ2119903

1198991198862

119899) + (1119899

2

ℎ2

1198862

119899)) and 119864119860

119899222

=

119874(11198992

1198862

119899) Therefore it follows from Assumptions 5 and 7

that

1198601198992

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A19)

for almost every (119909 119911) Now consider 1198601198993(119909 119911) Write

1198601198993

(119909 119911) = 11986011989931

(119909 119911) + 11986011989932

(119909 119911) (A20)

where 11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ119892(119909 119911) + 119863119899(119909 119911) and

11986011989932

(119909 119911) = minus(+

+ 119886119899119868)

minus1

Δ(1198601198991

minus 119892)(119909 119911) It follows from(A11)-(A16) and (A20) that

11986011989932

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A21)

for almost every (119909 119911)To analyze 119860

11989931(119909 119911) define

1198611198991

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

1198611198992

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119909 119908) 119892 (119909 119909) 119889119909 119889119911 119889119908

1198611198993

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)

119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989911

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989912

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989921

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989922

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

(A22)

10 Mathematical Problems in Engineering

Define 120575 = ℎ2119903

+ (119899ℎ)minus1 HH show that

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

(11986111989921

+ 11986111989922

) (119909 119911)

(A23)

Define

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

11986111989921

(A24)

Then

119864100381710038171003817100381711986011989931

10038171003817100381710038172

le const [1198641003817100381710038171003817100381711986011989931

10038171003817100381710038171003817

2

+ 11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

]

(A25)

11986410038171003817100381710038171003817119860

11989931

10038171003817100381710038171003817

2

le const (1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

11986111989912

1003817100381710038171003817

4

+1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

(A26)

HH show that

1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817 = 119874(ℎ119903

119886119899

) (A27)

(1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

)12

= 119874(120575ℎ

2119903

119886119899

) (A28)

(1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

= 119874(1205752

1198862119899

) (A29)

See (611) (613) (614) and (615) of HH [11] Moreover

11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

= 119874(ℎ2119903minus1

1198991198862+(120572+1)120572

119899

+1

1198993ℎ51198864119899

+ℎ4119903

119899ℎ1198862119899

)

(A30)

See the arguments leading to (624) in HH [11] and theanalogous result for their equation (624) in HH [11] andthe analogous result for their quantity 119864119867

11989922 Combining

(A25)ndash(A30) with Assumptions 5 and 7 yields the result that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

=minus(119868 + 119879

+

Δ)minus1

119879+

11986111989912

radic119881119899(119909 119911)

+ 119900119901(1) (A31)

Now consider minus(119868 +119879+

Δ)minus1

119879+

11986111989912

Standard calculations forkernel estimators show that

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 119899minus1

119899

sum

119894=1

119891XZW (119909 119911W119894) 119892 (X

119894Z

119894) + 119874 (ℎ

119903

)

(A32)

Therefore

119879+

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 1198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

)

(A33)

119879+

11986111989912

(119909 119911) = 1198601198991

(119909 119911) minus 1198641198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

) (A34)

But

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911)

= 119879+

11986111989912

+ [(119868 + 119879+

Δ)minus1

minus 119868]119879+

11986111989912

= 119879+

11986111989912

+ ( + 119886119899119868)

minus1

Δ119879+

11986111989912

(A35)

Therefore it follows by combining Assumption 7 and equa-tions (A11) (A17) and (A34) that

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911) = 1198601198991

(119911) minus 1198641198601198991

(119909 119911) + 119903119899

(A36)

where 1198641199031198992

radic119881119899(119909 119911) = 119900(1) for almost every (119909 119911)

Combining this result with (A21) and (A31) gives

1198601198993

(119909 119911)

radic119881119899(119909 119911)

=minus [119860

1198991(119909 119911) minus 119864119860

1198991(119909 119911)]

radic119881119899(119909 119911)

+ 119900119901(1) (A37)

for almost every (119909 119911)Now consider 119860

1198994(119909 119911) HH show that

1198601198994

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ (1198601198992

minus 119879+

1198611198992) (119909 119911)

(A38)

Therefore it follows from (A19) and (A30) that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A39)

for almost every (119909 119911)Now combine (A19) (A37) and (A39) to obtain

1198781198992

(119909 119911)

radic119881119899(119909 119911)

=sum4

119895=1119860

119899119895(119909 119911)

radic119881119899(119909 119911)

=119864119860

1198991(119909 119911)

radic119881119899(119909 119911)

+ 119900119901(1)

(A40)

for almost every (119909 119911)The lemma follows by combining thisresult with (A16)

This completes the proof

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 no 1 pp 141ndash1831973

[3] F Antonelli ldquoBackward-forward stochastic differential equa-tionsrdquo The Annals of Applied Probability vol 3 no 3 pp 777ndash793 1993

[4] HWangW Li and XWang ldquoAsymptotic stabilization by statefeedback for a class of stochastic nonlinear systems with time-varying coefficientsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 258093 6 pages 2014

[5] W Zhang and G Li ldquoDiscrete-time indefinite stochastic linearquadratic optimal control with second moment constraintsrdquoMathematical Problems in Engineering vol 2014 Article ID278142 9 pages 2014

[6] Y Su and L Lin ldquoSemi-parametric estimation for forward-backward stochastic differential equationsrdquo Communications inStatistics Theory and Methods vol 38 no 11 pp 1759ndash17752009

[7] X Chen and L Lin ldquoNonparametric estimation for FBS-DEs models with applications in financerdquo Communications inStatisticsmdashTheory and Methods vol 39 no 14 pp 2492ndash25142010

[8] L Lin F Li and L X Zhu ldquoOn regressionwith variance built-inmean regression function a new financial modelrdquo Manuscript2009

[9] Q Zhang and L Lin ldquoTerminal-dependent statistical inferencesfor FBSDErdquo Stochastic Analysis and Applications vol 32 pp128ndash151 2014

[10] C Hsiao Analysis of Panel Data vol 36 of Econometric SocietyMonographs Cambridge University Press Cambridge UK 2ndedition 2003

[11] P Hall and J L Horowitz ldquoNonparametric methods for infer-ence in the presence of instrumental variablesrdquo The Annals ofStatistics vol 33 no 6 pp 2904ndash2929 2005

[12] A N Kolmogorov and U A Rozanov ldquoOn the strong mixingconditions of a stationary Gaussian processrdquo Theory of Proba-bility and Its Applications vol 2 pp 222ndash227 1960

[13] R C Bradley and W Bryc ldquoMultilinear forms and measures ofdependence between random variablesrdquo Journal of MultivariateAnalysis vol 16 no 3 pp 335ndash367 1985

[14] C R Lu and Z Y Lin Limit Theories for Mixing DependentVariables Science Press Beijing China 1997

[15] D N Politis and J P Romano ldquoA general resampling scheme fortriangular arrays of120572-mixing randomvariableswith applicationto the problem of spectral density estimationrdquo The Annals ofStatistics vol 20 no 4 pp 1985ndash2007 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

of parametric and nonparametric methods to deal withestimation and test for the OSDE However these methodscannot be directly employed to infer the BSDE and FBSDEbecause the two models are related to a terminal conditionForward-backward stochastic differential equations are usedin biology systems mathematical finance insurance realestate multiagent and network control See Antonelli [3]Wang et al [4] Zhang and Li [5] and so on

For the FBSDE defined above the statistical inference wasinvestigated initially by Su and Lin [6] and Chen and Lin [7]Furthermore by financial and ecological problems a relevantstatistical model was proposed by Lin et al [8] Howeverthey did not take the terminal condition into account inthe inference procedure In the framework of the FBSDEmentioned above the terminal condition is additional whichis not nested into the equation Thus there is an essentialdifficulty to use the terminal condition to refine the inferenceprocedure

As a result their methods fail to cover the full problemsgiven in the FBSDE Zhang and Lin [9] proposed twoterminal-dependent estimationmethods via terminal controlvariable for the integral form models of FBSDE Howeverthey only considered the parametric form of the generator 119892in their paper

This paper intends to explore the method to fulfill theterminal-dependent inference quasi-instrumental variablemethods It is worth mentioning that the key point of ourmethod is the use of the terminal condition informationrather than neglecting it This change leads to a completelynew work among the existing researches The key techniquein ourmethod is the use of quasi-instrumental variable whichis similar but not the same as instrumental variable (IV) It isknown that IV is widely employed in applied econometrics toachieve identification and carry out estimation and inferencein the model containing endogenous explanatory variablesor panel data see Hsiao [10] for an overview of the relevantstatistical inference and econometric interpretation and seeHall and Horowitz [11] for recent work on nonparametricinstrumental variable estimation

Through the backward equation (2) of FBSDE we get aregression model To use the terminal condition informa-tion we put the terminal condition as a quasi-instrumentalvariable and introduce it into our model However when aconstraint is appended artificially the original model maychange to be biased in the sense of 119864(119885

119904119889119861

119904| 119883

119904 120585) = 0

because the constraint condition influences the increase trendof wealth so that 119885

119904119889119861

119904may deviate from the original center

zero in other words due to the constraint the trajectoryof 119884

119904may departure from the original expectation so that

119885119904119889119861

119904cannot be regarded as errorTherefore some problems

arise naturally including how to correct the bias of the modeland how to construct the constraint-dependent estimationTo solve these problems we will use remodeling method todraw terminal condition into differential equation similarbut not the same as IV called quasi-instrumental variablemethods in other words the terminal condition 120585 enters intothe equation as a control variable This remodeling methodtakes advantage of the terminal information naturally and theestimator performs quite well

We use the nonparametric form of the generator 119892 inmodel (2) because the correct FBSDEs model for a specifictopic can neither be provided automatically by financialmarket nor be derived from theory of mathematical financeand in lack of prior information about the structure ofa model nonparametric inference can provide a flexibleas well as robust description of a data-generating processEven in some cases when parametric models are availablenonparametricmethods are still employed to avoid themodelmisspecification that may lead to large errors in optionpricing and other problems from financial market So weadopt the nonparametric form that can endow the model (2)with flexibility and robustness

Note that 119885119904is usually unobservable and 119892 cannot be

completely specified in the financial marketThe problems ofinterest are therefore to give both proper estimations of thegenerator 119892 and the process 119885

119904based on the observed data

(119883119904 119884

119904) and the terminal expectation 120585

The remainder of the paper is organized as follows InSection 2 the FBSDE is rebuilt as a nonparametricmodel thatcontains the terminal condition as a quasi-instrumental vari-able Consequently a terminal-dependent estimation proce-dure is proposed Next we discuss the asymptotic propertiesof the newly proposed estimations in Section 3 Simulationstudy is proposed in Section 4 to illustrate our methods Theproofs of the theorems are presented in Appendix

2 Model and Method

In this section we propose a nonparametric estimator withthe help of quasi-instrumental variable

21 Model and Its Statistical Version We begin the followingoriginal model by combining (2)-(3)

119889119884119904= minus119892 (119904 119883

119904 119884

119904 119885

119904) 119889119904 + 119885

119904119889119861

119904 119884

119879= 120585

119889119883119904= 120583 (119904 119883

119904) 119889119905 + 120590 (119904 119883

119904) 119889119861

119904 119904 isin [119905 119879]

(4)

where 119861119905is the standard Brownian motion and 120585 is a R-

valued Borel function Here the generator 119892 is a function of119904 119883

119904 119884

119904 and 119885

119904 For the FBSDEs model (4) only one of

the backward components 119884119904 and the forward components

119883119904 can be observed Another backward component 119885

119904is

totally unobservable Furthermore the adapted process 119885119904

and terminal condition could be indicated as a function of119883119904In this section we present the statistical structure of

FBSDEs by taking advantage of quasi-instrumental variableand obtain the consistent asymptotically normal estimatorsof 119892 and 119885

119904based on observed data 119883

119904 119884

119904 and the terminal

condition 120585

22 Remodeling for Model (4) To construct terminal-dependent estimation for the generator 119892 and process 119885

119904

the key technique is how to plug the terminal condition intothe equation When 120585 is plugged into the model we call itthe quasi-IV similar but not the same as IV Evidently theproperty of Brownian motion shows that 119864(119885

119904119889119861

119904| 119883

119904) = 0

Mathematical Problems in Engineering 3

but 119864(119885119904119889119861

119904| 119883

119904 120585) = 0 which means drawing the terminal

control directly into the equation as the condition should notbe encouraged at the cost of model bias Rewriting the firstequation of (4) enables us to construct an unbiased model

119889119884119904= minus119892 (119904 119883

119904 119884

119904 119885

119904) 119889119904 + 119898 (119883

119904 120585) + 119880

119904 (5)

where 119898(119883119904 120585) = 119864(119885

119904119889119861

119904| 119883

119904 120585) 119880

119904= 119885

119904119889119861

119904minus

119898(119883119904 120585) and 119864(119880

119904| 119883

119904 120585) = 0 The newly defined

model (5) together with the second equation in (4) can bethought of as a quasi-IV FBSDE Because the equation in(5) contains the terminal condition 120585 we can construct theterminal-dependent estimation From the above definitionswe see that by bias correction the original model changesto be an additive nonparametric model with nonparametriccomponents minus119892(119904 119883

119904 119884

119904 119885

119904)119889119904 and 119898(119883

119904 120585) It shows that

when terminal condition is regarded as a quasi-IV and thenappended to the model the result model is unbiased andchanges to be nonparametric additive model

23 Estimation for 119885119904 Before estimating the model function

119898(119909119904 120585) and the generator 119892 we need to estimate 119885

119904firstly

because 119885119904is unobservable and it will be seen that the

estimators of the model function 119898(119909119904 120585) and the generator

119892 depend on 119885119904 Since the distribution of 120585 is supposed to

be known let 120585119894 1 le 119894 le 119896 for 119896 ge 1Δ be a sample of

120585 Suppose that for each terminal data 120585119895and equally spaced

time points 119904119894= 119904

1+(119894minus1)Δ 119894 = 1 119899 sube [0 119879] we record

the observed time series data119883

119904119894 119895 119884

119904119894 119895 119894 = 1 119899 119895 = 1 119896

= 119883119894119895 119884

119894119895 119894 = 1 119899 119895 = 1 119896

(6)

At any time point 119904 isin [119905 119879] 119885119905119909

119904 denoting 119885

119904and satisfying

the initial condition (119905 119909) is a determined function of 119883119905119909

119904

As was shown by Su and Lin [6] and Chen and Lin [7] wecan adopt a difference-based method to approximate 1198852 as

(119885119905119883119905

119904)2

=1

Δ119864(119884

119905+Δ119883119905+Δ

119904+Δminus 119884

119905119883119905

119904| 119883

119905 119905)

2

+ 119874 (Δ) (7)

It shows that the numerical approximation error to 1198852

119905

converges to zero at rate of order 119874119901(Δ)

For each 120585119895 if119885

119905depends on 119905 only via variable119883

119905 by (7)

and N-W kernel estimation method we estimate 1198852

119905at 119909

0by

1198852

1199090 119895=

sum119899minus1

119894=1Δminus1

(119884119894+1119895

minus 119884119894119895)2

119870ℎ119883

(119883119894119895

minus 1199090)

sum119899minus1

119894=1119870ℎ119883

(119883119894119895

minus 1199090)

(8)

Otherwise we estimate 1198852

119905at (119909

0 1199050) by

1198852

1199090 1199050 119895

=sum119899minus1

119894=1Δminus1

(119884119894+1119895

minus 119884119894119895)2

119870ℎ119883

(119883119894119895

minus 1199090)119870

ℎ119905(119905119894minus 119905

0)

sum119899minus1

119894=1119870ℎ119883

(119883119894119895

minus 1199090)119870

ℎ119905(119905119894minus 119905

0)

(9)

where 119870ℎ119883

= 119870(sdotℎ119883)ℎ

119883and 119870

ℎ119905= 119870(sdotℎ

119905)ℎ

119905 119870(sdot) are reg-

ular kernel functions with ℎ119883and ℎ

119905being the corresponding

bandwidths

24 Estimation for 119898(119883119904120585) After plugging the estimator 119885

119904

into model (5) we still need to consider inference of119898(119909119904 120585)

As we all know the nonparametric function 119898(119883119904 120585) in (5)

can be acquired as 119898(119883119904 120585) = 119864(119889119884

119904+ 119892(119904 119883

119904 119884

119904 119885

119904)119889119904 |

119883119904 120585) We note that 119892(119904 119883

119904 119884

119904 119885

119904)119889119904 is a higher order

infinitesimal of 119885119904119889119861

119904when Δ tends to zero Under this

situation if 119892(119904 119883119904 119884

119904 119885

119904)119889119904 is ignored then

119898(119883119904 120585) ≐ 119864 (119889119884

119904| 119883

119904 120585) (10)

It implies that we can use ordinary nonparametric methodto estimate function 119898 For example we use the N-Wordinary nonparametric method to estimate119898(119883

119904 120585) valued

at (1199090 120585

0)

(1199090 120585

0)

=

sum119899minus1

119894=1sum119898

119895=1(119884

119894+1119895minus 119884

119894119895)119870

ℎ119883(119883

119894119895minus 119909

0)119870

ℎ120585(120585

119894119895minus 120585

0)

sum119899minus1

119894=1sum119898

119895=1119870ℎ119883

(119883119894119895

minus 1199090)119870

ℎ120585(120585

119894119895minus 120585

0)

(11)

where 119870ℎ119883

= 119870(sdotℎ119883)ℎ

119883and 119870

ℎ120585= 119870(sdotℎ

120585)ℎ

120585are regular

kernel functions with ℎ119883and ℎ

120585being the corresponding

bandwidths

25 Estimation for Generator 119892 As was shown in the non-parametric instrumental variables estimator of Hall andHorowitz [11] (hereinafter HH) we can adopt a nonpara-metric quasi-instrumental variables estimation to estimatethe generator 119892 So in the section we summarize the HHestimator of 119892 in the model

119864 [minus119889119884119905minus 119898 (119883

119905 120585) | 119883

119905 120585] = 119864 [119892 (119905 119883

119905 119884

119905 119885

119905) 119889119905 | 119883

119905 120585]

(12)

Since (1199090 120585

0) and 119885

2

1199090 119895are the consistent estimator of

119898(1199090 120585

0) and 119885

2

1199090 119895 respectively we use them instead of

119898(119883119904 120585) and 119885

119904in the above model and we get

119864 [minus119889119884119904minus (119883

119904 120585) | 119883

119904 120585]

= 119864 [119892 (119904 119883119904 119884

119904 119885

119904) 119889119905 | 119883

119904 120585]

(13)

Because 119885119904is function of 119883

119904and 119884

119904 for simplicity of

presentation we denote 119892(119904 119883119904 119884

119904 119885

119904) = 119892(119883

119904 119884

119904)Thus the

model becomes

119864 [minus119889119884119904minus (119883

119904 120585) | 119883

119904 120585] = 119864 [119892 (119883

119904 119884

119904) 119889119905 | 119883

119904 120585]

(14)

Let Y119894= ((119884

119894+Δminus 119884

119894) minus (119883

119894 120585))Δ X

119894= 119883

119894 Z

119894= 119884

119894

W = 120585 and U119894= 119881

119894radicΔ the model becomes

Y119894= 119892 (X

119894Z

119894) + U

119894 119864 (U

119894| X

119894W

119894) = 0 (15)

It is assumed that the support of (XZW) is containedin [0 1]

3 This assumption can always be satisfied by ifnecessary carrying outmonotone increasing transformationsofXZ andW For example one can replaceXZ andW by

4 Mathematical Problems in Engineering

Φ(X) Φ(Z) and Φ(W) where Φ is the normal distributionfunction We take (Y XZWU) to be a vector where Y

and U are scalars X and W are supported on [0 1] and Z

is supported on [0 1] The model is

Y119894= 119892 (X

119894Z

119894) + U

119894 119864 (U

119894| Z

119894W

119894) = 0 (16)

where (Y119894X

119894Z

119894W

119894U

119894) for 119894 ge 1 are independent and

identically distributed as (Y XZWU) Thus X and Z areendogenous and exogenous explanatory variables respec-tively Data (Y

119894X

119894Z

119894W

119894U

119894) for 1 le 119894 le 119899 are observed

Let119891XZW denote the density of (XZW) write119891Z for thedensity of Z and for each 119909

1 119909

2isin [0 1]

119901 and put

119905119911(119909

1 119909

2) = int119891XZW (119909

1 119911 119908) 119891XZW (119909

2 119911 119908) 119889119908 (17)

Define the operator 119879119911on 119871

2[0 1]

119901 by

(119879119911120595) (119909) = int 119905

119911(120585 119909) 120595 (120585) 119889120585 (18)

It may be proved that for each 119911 for which 119879minus1

119911exists

119892 (119909 119911)

= 119891Z (119911) 119864W|Z

times 119864 (Y | Z = 119911W) (119879minus1

119911119891XZW) (119909 119911W) | Z = 119911

(19)

where 119864W|Z denotes the expectation with respect to thedistribution of W conditional on Z In this formulation(119879

minus1

119911119891XZW)(119909 119911W) denoted the result of applying 119879minus1

119911to the

function 119891XZW(sdot 119911W) and evaluating the resulting functionat 119909

To construct an estimator of 119892(119909 119911) given ℎ gt 0 and 119909 =

119909(1) and 120585 = 120585

(1) let 119870ℎ(119909 120585) = 119870

ℎ(119909

(119895)

120585(119895)

) put 119870ℎ(119911 120585)

analogously for 119911 and 120585 let ℎ119909 ℎ

119911gt 0 and define

119891XZW (119909 119911 119908)

=1

119899ℎ2119909ℎ119911

119899

sum

119894=1

119870ℎ119909

(119909 minusX119894 119909)119870

ℎ119911(119911 minus Z

119894 119911)119870

ℎ119909(119908 minusW

119894 119908)

119891minus119894

XZW (119909 119911 119908)

=1

(119899 minus 1) ℎ2119909ℎ119911

sum

1le119895le119899119895 = 119894

119870ℎ119909

(119909 minusX119895 119909)

times 119870ℎ119911

(119911 minus Z119895 119911)119870

ℎ119911(119908 minusW

119895 119908)

119911(119909

1 119909

2) = int119891XZW (119909

1 119911 119908) 119891XZW (119909

2 119911 119908) 119889119908

(119911120595) (119909 119911 119908) = int

119911(120585 119909) 120595 (120585 119911 119908) 119889120585

(20)

where120595 is a function from1198773 to a real lineThen the estimator

of 119892(119909 119911) is

119892 (119909 119911) =1

119899

119899

sum

119894=1

(+

119911119891minus119894

XZW) (119909 119911W119894) 119884

119894119870ℎ119911

(119911 minus Z119894 119911) (21)

3 Asymptotic Results

In this section we study the asymptotic properties of ourproposed estimators All proofs are presented in Appendix

31 Asymptotic results of119885119904 To give the asymptotic results of

119885119904 we need the following conditions

(a) 1198831 119883

119899are 120588-mixing dependent namely the 120588-

mixing coefficients 120588(119897) satisfy 120588(119897) rarr 0 as 119897 rarr infinwhere

120588 (119897) = sup119864(119883119894+119897119883119894)minus119864(119883119894+119897)119864(119883119894) = 0

1003816100381610038161003816119864 (119883119894+119897119883119894) minus 119864 (119883

119894+119897) 119864 (119883

119894)1003816100381610038161003816

radicVar (119883119894+119897)Var (119883

119894)

(22)

with119883119894= 119883(119905

119894)

(b) |119885119894| le 119862 (a s) uniformly for 119894 = 1 119899 where 119862 is a

positive constant and 119885119894= 119885(119905

119894)

(c) The continuous kernel function 119870(sdot) is symmetricabout 0 with a support of interval [minus1 1] and

int

1

minus1

119870 (119906) 119889119906 = 1 1205902

119870= int

1

minus1

1199062

119870 (119906) 119889119906 = 0

int

1

minus1

|119906|119895

119870119896

(119906) 119889119906 lt infin for 119895 le 119896 = 1 2

(23)

Condition (a) is commonly used for weakly dependentprocess see for example Kolmogorov and Rozanov [12]Bradley and Bryc [13] Lu and Lin [14] and Su and Lin [6]Condition (b) is also reasonable because as is shown by (10)119885119905can be regarded as the deviation between the adjacent two

observations Condition (c) is standard for nonparametrickernel function

Theorem 1 Besides conditions (a) (b) and (c) let119883

1 119883

119899 be an observation sequence on a stationary

120588-mixing Markov process with the 120588-mixing coefficientssatisfying 120588(119897) = 120588

119897 for 0 lt 120588 lt 1 Furthermore 1198831 119883

119899

have a common and probability density 119901(119909) and for eachinterior point 119909

0in the support of 119901(sdot) 119901(119909

0) gt 0 1198852

(1199090) gt 0

the functions 119901(119909) and 119885(119909) have continuous two derivativesin neighborhood of 119909

0 As 119899 rarr infin such that 119899ℎ rarr infin

119899ℎ5

rarr 0 and 119899ℎΔ2

rarr 0 then

radic119899ℎ (1198852

(1199090) minus 119885

2

(1199090))

119889

997888rarr (01198854

(1199090) 119869

119870

119901 (1199090)

) (24)

where 119869119870= int

1

minus1

1198702

(119906)119889119906 lt infin

The asymptotic result in Theorem 1 is standard fornonparametric kernel estimator and here undersmoothing isused to eliminate asymptotic bias

32 Asymptotic results of 119892(119909119911) This section gives con-ditions under which the HH estimator of the generator

Mathematical Problems in Engineering 5

119892 is asymptotically distributed as 119873(0 119868) The followingadditional notations are used

Define U119894

= Y119894

minus 119892(X119894Z

119894) 119878

1198991(119909 119911) =

119899minus1

sum119899

119894=1U119894+

119891(minus119894)

XZW(119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911) and 119878

1198992(119909 119911) =

119899minus1

sum119899

119894=1119892(X

119894Z

119894)

+

119891(minus119894)

XZW(119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911) Then

119892(119909 119911) = 1198781198991(119909 119911) + 119878

1198992(119909 119911) Define 119879+

= (119879+ 119886119899119868)

minus1 Write

1198781198991

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894(119879

+

119891XZW) (119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911)

+ 119899minus1

119899

sum

119894=1

U119894(

+

119891(minus119894)

XZW minus 119879+

119891XZW)

times (119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911)

= 11987811989911

(119909 119911) + 11987811989912

(119909 119911)

(25)

Define 119881119899(119909 119911) = 119899

minus1 Var[U(119879+

119891XZW)(119909 119911W)] It followsfrom a triangular array version of the Lindeberg-Levy centrallimit theorem that 119878

11989911(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0 1) as 119899 rarr

infin Therefore [119892(119909 119911) minus 119892(119909 119911)]radic119881119899(119909 119911)rarr

119889

119873(0 1) if[119878

11989912(119909 119911) + 119878

1198992(119909 119911) minus 119892(119909 119911)]radic119881

119899(119909 119911) = 119900

119901(1)

Assumption 2 The data Y119894X

119894 Z

119894W

119894are independently and

identically distributed as (Y XXW) where (XZW) issupported on [0 1]

3 and 119864[Y minus 119892(XZ) | WZ] = 0

Assumption 3 The distribution of (XZW) has a density119891XZW with respect to Lebesgue measure Moreover 119891XZW is119903 times differentiable with respect to any combination of itsarguments where derivatives at the boundary of [0 1]3 aredefined as one sided derivativesThe derivatives are boundedin absolute value by 119862 In addition 119892 is 119903 times differentiableon [0 1]

2 with derivatives at 0 and 1 defined as one sidedThe derivatives of 119892 are bounded in absolute value by 119862 Inaddition 119864[Y 2

| XZW] le 119862 and 119864[Y 2

| XZW] le 119862 and119864[U2

| ZW] ge 119862119880for some finite constant 119862

119880

Assumption 4 The constants 120572 and 120573 satisfy 120572 gt 1 120573 gt 12and 120573 minus 12 le 120572 lt 2120573 Moreover 119887

119895le 119862119895

minus120573 119895minus120572 le 119862120582119895 and

suminfin

119896=1|119889

119911119895119896| le 119862119895

minus1205722 for all 119895 ge 1 In addition there are finitestrictly positive constants 119862

1205821and 119862

1205822 such that 119862

1205821le 120582

119895le

1198621205822119895minus120572 for all 119895 ge 1

Assumption 5 The tuning parameters 119886119899and ℎ satisfy 119886

119899≍

119899minus(120588120572)(2120573+120572) and ℎ ≍ 119899

minus1 where 119903 isin [1198601015840

2 119860

1015840

3]

Assumption 6 119870ℎdenotes a generalized kernel function with

the properties 119870ℎ(119906 119905) = 0 if 119906 gt 119905 or 119906 lt 119905 minus 1 for all

119905 isin [0 1]ℎminus(119895+1)

int119905minus1

119905119906119895

119870ℎ(119906 119905)119889119906 = 1 if 119895 = 0 else 0 if

1 le 119895 le 119903 minus 1 For each 120585 isin [0 1] 119870ℎ(ℎ 120585) is supported

on [(120585 minus 1)ℎ 120585ℎ] cap 120581 where 120581 is a compact interval notdepending on 120585 Moreover

supℎgt0120585isin[01]119906isin120581

119870ℎ(ℎ119906 120585) |lt infin (26)

Assumption 7 Consider 119864W[119879+

119891XZW(119909 119911W)]2

≍

119864W[119879+

119891XZW(sdot sdotW)]2 and 119864W[119879

+

119891XZW(sdot sdotW)]2

≍

int1

0

119879+

119891XZW(sdot sdotW)2

119889119908

Theorem 8 Let Assumptions 2ndash7 hold Then

119892 (119909 119911) minus 119892 (119909 119911)

radic119881119899(119909 119911)

997888rarr119889

119873(0 119868) (27)

holds except possibly on a set of (119909 119911) values whose Lebesgueis 0

Corollary 9 Let Assumptions 2ndash7 hold And if 119881119899(119909 119911) is

replaced with the consistent estimator

119899(119909 119911) = 119899

minus1

119899

sum

119894=1

U2

119894[

+

119891minus119894

119909119908(119911W

119894)119870

119902ℎ119911(119911 minus Z

119894 119911)]

2

(28)

where U119894= Y

119894minus 119892(X

119894Z

119894) This yields the Studentized statistic

[119892(119909 119911) minus 119892(119909 119911)]radic119899(119909 119911) Then

119892 (119909 119911) minus 119892 (119909 119911)

radic119899(119909 119911)

997888rarr119889

119873(0 119868) (29)

holds except possibly on a set of (119909 119911) values whose Lebesgueis 0

As was shown in the remark given in the previoussection even the conditional mean of error of the model isnonzero and the newly proposed estimation is consistentbecause of themixing dependency for details see the proof ofTheorem 8 Furthermore because of the terminal conditionthe asymptotic variance is larger than that without the use ofthe terminal condition

4 Simulation Studies

In this section we investigate the finite-sample behaviors bysimulation

Example 10 We consider a simple FBSDE as

119889119884119905= (

120583 minus 119903

120590119885119905+ 119903119884

119905)119889119905 + 119885

119905119889119861

119905

≜ (119887119884119905+ 119888119885

119905) + 119885

119905119889119861

119905 119884

119879= 120585

(30)

where119883119905is Geometric Brownian motion for modeling stock

price satisfying

119889119883119905= 120583119883

119905119889119905 + 120590119883

119905119889119861

119905 119883

0= 119909 (31)

while the riskless asset is the same as formula (31) 119889119875119905

=

1199031198750119889119905

Firstly let 120583 = 01 120590 = 001 Δ = 012 119899 = 300119879 = 366 and 119899

0= 119899

1= 10 Obviously119885

119905= 119899

1120590119883

119905We adopt

Epanechnikov kernel defined by119870(119906) = 34(1minus1199062

)119868(|119906| le 1)

6 Mathematical Problems in Engineering

012

01

008

006

004

002

0

Curve of ZEstimated curve of Z

0 5 10 15 20 25 30 35 40

(a)

Curve of gEstimated curve of g

0 5 10 15 20 25 30 35 40

14

12

1

08

06

04

02

0

(b)

Figure 1 The real lines are the true curves of 119885119905and function 119892(119905) respectively and the dashed ones are estimated curves for them in

Example 10

where 119868(sdot) is the indicator function For bandwidth selectionvarious data-driven techniques have been developed suchas cross-validation the plug-in method and the empiricalbias method However these useful tools require additionalcomputation intensiveness In our simulation we simplyapply the rule of thumb bandwidth selector For bandwidthselection bandwidth ℎ = std(119909)119899minus15 The values of thetuning parameters are 119886

119899= 005 120572 = 12 120573 = 1 Figure 1

presents the estimated curves for diffusion 119885119905and drift 119892 by

one simulation

Example 11 According to the theory ofmathematical financewe represent a European call option by the following FBSDEsmodel

119889119883119904= 119887119883

119904119889119904 + 120590119883

119904119889119882

119904

119889119884119904= [119903119884

119904+ (119887 minus 119903) 120590

minus1

119885119904] 119889119904 + 119885

119904119889119882

119904

1198830= 119909 119884

119879= (119883

119879minus 119870)

+

119904 isin [0 119879]

(32)

Here 1198831199040le119904le119879

and 1198841199040le119904le119879

are the price processes of thestock and the option respectively and119870 is the striking priceat the expiration date 119879 119883

1199040le119904le119879

follows the geometricBrownian motion as

119889119883119904= 119887119883

119904119889119904 + 120590119883

119904119889119882

119904

1198830= 119909 119904 isin [0 119879]

(33)

We use the Euler scheme to generate the price series ofthe stock as

119883119894+1

minus 119883119894= 119887119883

119894Δ + 120590119883

119894Δ12

120598119894 119894 = 0 119899 minus 1 (34)

where 120598119894119899minus1

119894=0is an iid series with standard normality

The price series by Black Scholes formula is part of thesolution of the FBSDEs above at discrete time points that is

119884119894= 119883

119894119873(119889

119894

+) minus 119890

minus119903(119899minus119894)Δ

119870119873(119889119894

minus) (35)

which together with

119885119894= 120590119883

119894119873(119889

119894

+) (36)

gives us data generating formulae where

119873(119910) =1

radic2120587int

119910

minusinfin

119890minus11990922

119889119909 (37)

is a cumulative normal function and

119889119894

plusmn=ln (119883

119894119870) + (119903 plusmn 120590

2

2) ((119899 minus 119894) Δ)

120590radic(119899 minus 119894) Δ (38)

We produce the true curve of the drift coefficient by

119892119894= minus119903119884

119894minus (119887 minus 119903) 120590

minus1

119885119894 (39)

We first apply formulas (21) and (11) to estimate 119892119894and

1198852

119894 respectively We adopt Epanechnikov kernel defined by

119870(119906) = 34(1 minus 1199062

)119868(|119906| le 1) where 119868(sdot) is the indicatorfunction For bandwidth selection we simply apply the ruleof thumb bandwidth selector

ℎ = constant times std (1198840 119884

119899minus1) 119899

minus15 (40)

to implement the estimationLet 119870 = 110 119883

0= 100 119887 = 01 120590 = 018 119903 = 008

119879 = 60 and Δ = 1100 The bandwidth parameters ℎ = 606

and ℎ = 067 are used for estimation of119892119904and119885

119904 respectively

The values of the tuning parameters are 119886119899= 005 120572 = 12

and 120573 = 1 To see the performance of our estimationmethodthe simulated and the estimated curves of the two coefficientsof the backward equation are displayed in Figures 2 and 3

Mathematical Problems in Engineering 7

0

0

10 20 30 40 50 60minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

minus100

Curve of gEstimated curve of g

Figure 2 The simulated curve and the estimated curves of 119892119904in

Example 11

00

10 20 30 40 50 60

Curve of ZEstimated curve of Z

800

700

600

500

400

300

200

100

Figure 3 The simulated curve and the estimated curves of 119885119904in

Example 11

Appendix

A Proofs

Proof of Theorem 1 Denote C = 1198831 119883

119899 By the

Taylor expansion and formula (8) we have

119864 (1198852

(1199090) | C)

=sum119899minus1

119894=1Δminus1

119870ℎ(119883

119894minus 119909

0) 119864 ((119884

119894+1minus 119884

119894)2

| C)

sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0)

=sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0) (119885

2

119894+ 119874 (Δ))

sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0)

=int119870

ℎ(119883

119894minus 119909

0) (119885

2

(119909)+119874 (Δ))119901 (119909) 119889119909 (1+ 119874119901(119899ℎ)

minus12

)

int119870ℎ(119883

119894minus 119909

0) 119901 (119909) 119889119909 (1+119874

119901(119899ℎ)

minus12

)

= ( (1198852

(1199090) + 119874 (Δ))

times (119901 (1199090) + (12) ℎ

2

119901(2)

(1199090) 120590

2

119870+ 119900 (ℎ

2

))

times (1 + 119874119901(119899ℎ)

minus12

) )

times ( (119901 (1199090) + (12) ℎ

2

119901(2)

(1199090) 120590

2

119870+ 119900 (ℎ

2

))

times (1 + 119874119901(119899ℎ)

minus12

) )

minus1

= 1198852

(1199090) +

119901(2)

(1199090)

2119901 (1199090)ℎ2

1198852

(1199090) 120590

2

119870+ 119900 (ℎ

2

) + 119874 (Δ)

(A1)

Furthermore

Var (1198852

(1199090) | C)

=1

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

times

119899minus1

sum

119894=1

Δminus2

1198702

ℎ(119883

119894minus 119909

0)Var ((119884

119894+1minus 119884

119894)2

| C)

+

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

Δminus2 cov (119870

ℎ(119883

119894minus 119909

0) (119884

119894+1minus 119884

119894)

119870ℎ(119883

119894+119896minus 119909

0) (119884

119894+119896+1minus 119884

119894+119896) | C)

(A2)

From the conditions of Markov process and 120588-mixing coeffi-cient1003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

Δminus2 cov ( 119870

ℎ(119883

119894minus 119909

0) (119884

119894+1minus 119884

119894)

119870ℎ(119883

119894+119896minus 119909

0) (119884

119894+119896+1minus 119884

119894+119896) )

1003816100381610038161003816100381610038161003816100381610038161003816

=1

(119899 minus 1)2

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

10038161003816100381610038161003816100381610038161003816119864 ((Δ)

minus2

(119884119894+1

minus 119884119894)2

(119884119894+119896+1

minus 119884119894+119896

)2

times (119870ℎ(119883

119894minus 119909

0) minus 119864 (119870

ℎ(119883

119894minus 119909

0)))

times(119870ℎ(119883

119894+119896minus119909

0)minus119864 (119870

ℎ(119883

119894+119896minus119909

0))))

10038161003816100381610038161003816100381610038161003816

=1

(119899 minus 1)2

10038161003816100381610038161003816100381610038161003816119864 (119885

2

1198941198852

119894+119897(119870

ℎ(119883

119894minus 119909

0) minus 119864 (119870

ℎ(119883

119894minus 119909

0)))

times (119870ℎ(119883

119894+119896minus 119909

0) minus 119864 (119870

ℎ(119883

119894+119896minus 119909

0))))

10038161003816100381610038161003816100381610038161003816

+ 119874 (Δ)

le119862

(119899 minus 1)2

ℎ

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

120588119896

= 119874(1

119899ℎ) = 119900 (1)

(A3)

8 Mathematical Problems in Engineering

Note that (119884119894+1

minus 119884119894)radicΔ = 119892(119905

119894 119884

119894 119885

119894)radicΔ + 119885

119894120578119894 where

119864(120578119894) = 0 Var(120578

119894) = 1 Thus Var((119884

119894+1minus119884

119894)radicΔ) = 119885

4

119894+119874(Δ)

and furthermore

Var (1198852

(1199090) | C)

=sum119899minus1

119894=1Δminus2

1198702

ℎ(119883

119894minus 119909

0)Var ((119884

119894+1minus 119884

119894)2

| C)

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

+ 119874119901(1)

=sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0) (119885

4

(1199090) + 119874 (radicΔ))

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

+ 119874119901(1)

=1198854

(1199090) 119869

119870+ 119874 (radicΔ)

119899ℎ119901 (1199090)

(1 + 119874119901(119899ℎ)

minus12

)

(A4)

To our interest both the conditional expectation and varianceare independent onC so the condition could be erased

From Lemma A1 of Politis and Romano [15] and therelation between the 120572-mixing condition and the 120588-mixingcondition (eg Theorem 111 of Lu and Lin [14]) we canensure that (119884

119894+1minus 119884

119894)2

119894 = 1 119899 minus 1 is a 120588-mixingdependent process and the mixing coefficient denoted by120588119884(119897) satisfies

infin

sum

119896=1

120588119884(2

119896

) le 119862

infin

sum

119896=1

120588 (2119896

) =

infin

sum

119896=1

1205882119896

lt infin (A5)

where119862 is a positive constant Finally we use the central limittheorems for 120588-mixing dependent process (eg Theorem401 of Lu and Lin [14]) to complete this proof

Proof of Theorem 8 Theorem 8 follows from proving that1198781198991(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0 1198682) and [119878

1198992(119909 119911) minus 119892(119909 119911)]

radic119881119899(119909 119911) = 119900

119901(1) except possibly if (119909 119911) belongs to a

set of Lebesgue measure 0 The first result is established inLemma A1 and the second is established in Lemma A2Throughout this Appendix ldquofor almost every (119909 119911)rdquo meansldquofor every (119909 119911) isin [0 1]

2 except possibly a set of Lebesguemeasure 0rdquo We make repeated use of the fact that if 1198641205952 =

119874(119899minus119904

) for some 119904 gt 0 then120595(119909 119911) = 119900119901(119899

minus119904

) for almost every(119909 119911)

Lemma A1 (asymptotic normality of 1198781198991(119909 119911)radic119881

119899(119909 119911))

Let Assumptions 2ndash7 hold Then 1198781198991(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0

1198682) for almost every (119909 119911)

Proof Define 11987811989911

(119909 119911) = 119899minus1

sum119899

119894=1U119894(119879

+

119891XZW)(119909 119911W119894)

1198601198992

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[119879

+

(119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

1198601198993

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[(

+

minus 119879+

) 119891XZW] (119909 119911W119894)

1198601198994

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[(

+

minus 119879+

) (119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

(A6)

Then 1198781198991(119909 119911) = 119878

11989911(119909 119911)+119860

1198992(119909 119911)+119860

1198993(119909 119911)+119860

1198994(119909 119911)

11987811989911

(119909 119911)radic119881119899(119909 119911)rarr

119889

119873(0 1198682) by a triangular array version

of the Lindeberg-Levy central limit theorem The proof ofthe triangular-array version of the theorem is identical to theproof of the ordinary Lindeberg-Levy theorem The lemmafollows if we can prove that 119860

119899119895(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for

119895 = 2 3 4 and almost every (119909 119911) isin [0 1]2

Assumption 7 and arguments like those leading to (62)of HH [11] show that

∬

1

0

119881119899(119909 119911) 119889119909 119889119911 ≍ 119899

minus[2120573+120572minus120588(120572+1)](2120573+120572)

(A7)

It follows from the Cauchy-Schwartz inequality 119864(119891(minus119894)

XZW minus

119891XZW) = 119874(ℎ1015840

) and Var(119891(minus119894)

XZW) = 119874[1(119899ℎ2

)] that

11986410038171003817100381710038171198601198992

10038171003817100381710038172

= 119874(1

1198992ℎ21198862119899

+ℎ2119903

1198991198862119899

) (A8)

Therefore it follows from Assumptions 5 and 7 that119860

1198992(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909 119911) Now

consider 1198601198993(119909 119911) Define the operator Δ = minus 119879 Then

1198601198993

(119909 119911) = minus ( + 119886119899119868) Δ119860

1198991(119909 119911) (A9)

Therefore the Cauchy-Schwartz inequality gives

11986410038171003817100381710038171198601198992

10038171003817100381710038172

le 11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

119864100381710038171003817100381711987811989911

10038171003817100381710038172

= 11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

∬

1

0

119881119899(119909 119911) 119889119909 119889119911

(A10)

HH show that

11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

= 119874(1

119899ℎ1198862119899

+ℎ2119903

1198862119899

) (A11)

Therefore it follows from Assumptions 5 and 7 that119860

1198993(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909 119911) Finally

some algebra shows that

1198601198994

(119909 119911) = minus( + 119886119899119868)

minus1

Δ1198601198992

(119909 119911) (A12)

Therefore 1198601198994(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909

119911) follows from (A11) and 1198601198992(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1)

Lemma A2 (asymptotic negligibility of 1198781198992(119909 119911) minus 119892(119909 119911))

Let Assumptions 2ndash7 hold Then 1198781198992(119909 119911) minus 119892(119909 119911)

radic119881119899(119909 119911) = 119900

119901(1) for almost every (119909 119911)

Mathematical Problems in Engineering 9

Proof Define

119863119899(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times (119891XZW minus 119891XZW) (119909 119911 119908) 119889120579 119889120578 119889119908

1198601198991

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894 119885

119894) (119879

+

119891XZW) (119909 119911W119894)

(A13)

Redefine

1198601198992

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

minus 119863119899(119909 119911)

1198601198993

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) 119891XZW] (119909 119911W119894) + 119863

119899(119909 119911)

1198601198994

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) (119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894)

(A14)

Then 1198781198992(119909 119911) = sum

4

119895=1119860

119899119895(119909 119911) Arguments identical to

those used to derive (62) and (63) of HH [11] show that119864119860

1198991minus 119892

2

= 119874[119899minus120588(21205731)(2120573+120572)] and

∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874119899minus[2120573+120572minus120588(120572+1)](2120573+120572)

(A15)

Therefore it follows from Assumptions 5 and 7 that

[1198641198601198991

(119909 119911) minus 119892 (119909 119911)]

radic119881119899(119909 119911)

= 119900 (1) (A16)

119881minus1

119899(119909 119911)∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874 (1) (A17)

for almost every (119909 119911)Now consider 119860

1198992(119909 119911) Define

119863119899119894(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times(119891(minus119894)

XZWminus119891XZW)(119909 119911 119908) 119889120579 119889120578 119889119908

11986011989921

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894) minus 119863

119899119894(119909 119911)

(A18)

and 11986011989922

(119909 119911) = 119899minus1

sum119899

119894=1[119863

119899119894(119909 119911) minus 119863

119899(119909 119911)] HH show

that 11986411986011989921

2

= 119874((ℎ2119903

1198991198862

119899) + (1119899

2

ℎ2

1198862

119899)) and 119864119860

119899222

=

119874(11198992

1198862

119899) Therefore it follows from Assumptions 5 and 7

that

1198601198992

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A19)

for almost every (119909 119911) Now consider 1198601198993(119909 119911) Write

1198601198993

(119909 119911) = 11986011989931

(119909 119911) + 11986011989932

(119909 119911) (A20)

where 11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ119892(119909 119911) + 119863119899(119909 119911) and

11986011989932

(119909 119911) = minus(+

+ 119886119899119868)

minus1

Δ(1198601198991

minus 119892)(119909 119911) It follows from(A11)-(A16) and (A20) that

11986011989932

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A21)

for almost every (119909 119911)To analyze 119860

11989931(119909 119911) define

1198611198991

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

1198611198992

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119909 119908) 119892 (119909 119909) 119889119909 119889119911 119889119908

1198611198993

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)

119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989911

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989912

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989921

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989922

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

(A22)

10 Mathematical Problems in Engineering

Define 120575 = ℎ2119903

+ (119899ℎ)minus1 HH show that

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

(11986111989921

+ 11986111989922

) (119909 119911)

(A23)

Define

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

11986111989921

(A24)

Then

119864100381710038171003817100381711986011989931

10038171003817100381710038172

le const [1198641003817100381710038171003817100381711986011989931

10038171003817100381710038171003817

2

+ 11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

]

(A25)

11986410038171003817100381710038171003817119860

11989931

10038171003817100381710038171003817

2

le const (1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

11986111989912

1003817100381710038171003817

4

+1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

(A26)

HH show that

1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817 = 119874(ℎ119903

119886119899

) (A27)

(1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

)12

= 119874(120575ℎ

2119903

119886119899

) (A28)

(1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

= 119874(1205752

1198862119899

) (A29)

See (611) (613) (614) and (615) of HH [11] Moreover

11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

= 119874(ℎ2119903minus1

1198991198862+(120572+1)120572

119899

+1

1198993ℎ51198864119899

+ℎ4119903

119899ℎ1198862119899

)

(A30)

See the arguments leading to (624) in HH [11] and theanalogous result for their equation (624) in HH [11] andthe analogous result for their quantity 119864119867

11989922 Combining

(A25)ndash(A30) with Assumptions 5 and 7 yields the result that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

=minus(119868 + 119879

+

Δ)minus1

119879+

11986111989912

radic119881119899(119909 119911)

+ 119900119901(1) (A31)

Now consider minus(119868 +119879+

Δ)minus1

119879+

11986111989912

Standard calculations forkernel estimators show that

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 119899minus1

119899

sum

119894=1

119891XZW (119909 119911W119894) 119892 (X

119894Z

119894) + 119874 (ℎ

119903

)

(A32)

Therefore

119879+

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 1198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

)

(A33)

119879+

11986111989912

(119909 119911) = 1198601198991

(119909 119911) minus 1198641198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

) (A34)

But

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911)

= 119879+

11986111989912

+ [(119868 + 119879+

Δ)minus1

minus 119868]119879+

11986111989912

= 119879+

11986111989912

+ ( + 119886119899119868)

minus1

Δ119879+

11986111989912

(A35)

Therefore it follows by combining Assumption 7 and equa-tions (A11) (A17) and (A34) that

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911) = 1198601198991

(119911) minus 1198641198601198991

(119909 119911) + 119903119899

(A36)

where 1198641199031198992

radic119881119899(119909 119911) = 119900(1) for almost every (119909 119911)

Combining this result with (A21) and (A31) gives

1198601198993

(119909 119911)

radic119881119899(119909 119911)

=minus [119860

1198991(119909 119911) minus 119864119860

1198991(119909 119911)]

radic119881119899(119909 119911)

+ 119900119901(1) (A37)

for almost every (119909 119911)Now consider 119860

1198994(119909 119911) HH show that

1198601198994

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ (1198601198992

minus 119879+

1198611198992) (119909 119911)

(A38)

Therefore it follows from (A19) and (A30) that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A39)

for almost every (119909 119911)Now combine (A19) (A37) and (A39) to obtain

1198781198992

(119909 119911)

radic119881119899(119909 119911)

=sum4

119895=1119860

119899119895(119909 119911)

radic119881119899(119909 119911)

=119864119860

1198991(119909 119911)

radic119881119899(119909 119911)

+ 119900119901(1)

(A40)

for almost every (119909 119911)The lemma follows by combining thisresult with (A16)

This completes the proof

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 no 1 pp 141ndash1831973

[3] F Antonelli ldquoBackward-forward stochastic differential equa-tionsrdquo The Annals of Applied Probability vol 3 no 3 pp 777ndash793 1993

[4] HWangW Li and XWang ldquoAsymptotic stabilization by statefeedback for a class of stochastic nonlinear systems with time-varying coefficientsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 258093 6 pages 2014

[5] W Zhang and G Li ldquoDiscrete-time indefinite stochastic linearquadratic optimal control with second moment constraintsrdquoMathematical Problems in Engineering vol 2014 Article ID278142 9 pages 2014

[6] Y Su and L Lin ldquoSemi-parametric estimation for forward-backward stochastic differential equationsrdquo Communications inStatistics Theory and Methods vol 38 no 11 pp 1759ndash17752009

[7] X Chen and L Lin ldquoNonparametric estimation for FBS-DEs models with applications in financerdquo Communications inStatisticsmdashTheory and Methods vol 39 no 14 pp 2492ndash25142010

[8] L Lin F Li and L X Zhu ldquoOn regressionwith variance built-inmean regression function a new financial modelrdquo Manuscript2009

[9] Q Zhang and L Lin ldquoTerminal-dependent statistical inferencesfor FBSDErdquo Stochastic Analysis and Applications vol 32 pp128ndash151 2014

[10] C Hsiao Analysis of Panel Data vol 36 of Econometric SocietyMonographs Cambridge University Press Cambridge UK 2ndedition 2003

[11] P Hall and J L Horowitz ldquoNonparametric methods for infer-ence in the presence of instrumental variablesrdquo The Annals ofStatistics vol 33 no 6 pp 2904ndash2929 2005

[12] A N Kolmogorov and U A Rozanov ldquoOn the strong mixingconditions of a stationary Gaussian processrdquo Theory of Proba-bility and Its Applications vol 2 pp 222ndash227 1960

[13] R C Bradley and W Bryc ldquoMultilinear forms and measures ofdependence between random variablesrdquo Journal of MultivariateAnalysis vol 16 no 3 pp 335ndash367 1985

[14] C R Lu and Z Y Lin Limit Theories for Mixing DependentVariables Science Press Beijing China 1997

[15] D N Politis and J P Romano ldquoA general resampling scheme fortriangular arrays of120572-mixing randomvariableswith applicationto the problem of spectral density estimationrdquo The Annals ofStatistics vol 20 no 4 pp 1985ndash2007 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

but 119864(119885119904119889119861

119904| 119883

119904 120585) = 0 which means drawing the terminal

control directly into the equation as the condition should notbe encouraged at the cost of model bias Rewriting the firstequation of (4) enables us to construct an unbiased model

119889119884119904= minus119892 (119904 119883

119904 119884

119904 119885

119904) 119889119904 + 119898 (119883

119904 120585) + 119880

119904 (5)

where 119898(119883119904 120585) = 119864(119885

119904119889119861

119904| 119883

119904 120585) 119880

119904= 119885

119904119889119861

119904minus

119898(119883119904 120585) and 119864(119880

119904| 119883

119904 120585) = 0 The newly defined

model (5) together with the second equation in (4) can bethought of as a quasi-IV FBSDE Because the equation in(5) contains the terminal condition 120585 we can construct theterminal-dependent estimation From the above definitionswe see that by bias correction the original model changesto be an additive nonparametric model with nonparametriccomponents minus119892(119904 119883

119904 119884

119904 119885

119904)119889119904 and 119898(119883

119904 120585) It shows that

when terminal condition is regarded as a quasi-IV and thenappended to the model the result model is unbiased andchanges to be nonparametric additive model

23 Estimation for 119885119904 Before estimating the model function

119898(119909119904 120585) and the generator 119892 we need to estimate 119885

119904firstly

because 119885119904is unobservable and it will be seen that the

estimators of the model function 119898(119909119904 120585) and the generator

119892 depend on 119885119904 Since the distribution of 120585 is supposed to

be known let 120585119894 1 le 119894 le 119896 for 119896 ge 1Δ be a sample of

120585 Suppose that for each terminal data 120585119895and equally spaced

time points 119904119894= 119904

1+(119894minus1)Δ 119894 = 1 119899 sube [0 119879] we record

the observed time series data119883

119904119894 119895 119884

119904119894 119895 119894 = 1 119899 119895 = 1 119896

= 119883119894119895 119884

119894119895 119894 = 1 119899 119895 = 1 119896

(6)

At any time point 119904 isin [119905 119879] 119885119905119909

119904 denoting 119885

119904and satisfying

the initial condition (119905 119909) is a determined function of 119883119905119909

119904

As was shown by Su and Lin [6] and Chen and Lin [7] wecan adopt a difference-based method to approximate 1198852 as

(119885119905119883119905

119904)2

=1

Δ119864(119884

119905+Δ119883119905+Δ

119904+Δminus 119884

119905119883119905

119904| 119883

119905 119905)

2

+ 119874 (Δ) (7)

It shows that the numerical approximation error to 1198852

119905

converges to zero at rate of order 119874119901(Δ)

For each 120585119895 if119885

119905depends on 119905 only via variable119883

119905 by (7)

and N-W kernel estimation method we estimate 1198852

119905at 119909

0by

1198852

1199090 119895=

sum119899minus1

119894=1Δminus1

(119884119894+1119895

minus 119884119894119895)2

119870ℎ119883

(119883119894119895

minus 1199090)

sum119899minus1

119894=1119870ℎ119883

(119883119894119895

minus 1199090)

(8)

Otherwise we estimate 1198852

119905at (119909

0 1199050) by

1198852

1199090 1199050 119895

=sum119899minus1

119894=1Δminus1

(119884119894+1119895

minus 119884119894119895)2

119870ℎ119883

(119883119894119895

minus 1199090)119870

ℎ119905(119905119894minus 119905

0)

sum119899minus1

119894=1119870ℎ119883

(119883119894119895

minus 1199090)119870

ℎ119905(119905119894minus 119905

0)

(9)

where 119870ℎ119883

= 119870(sdotℎ119883)ℎ

119883and 119870

ℎ119905= 119870(sdotℎ

119905)ℎ

119905 119870(sdot) are reg-

ular kernel functions with ℎ119883and ℎ

119905being the corresponding

bandwidths

24 Estimation for 119898(119883119904120585) After plugging the estimator 119885

119904

into model (5) we still need to consider inference of119898(119909119904 120585)

As we all know the nonparametric function 119898(119883119904 120585) in (5)

can be acquired as 119898(119883119904 120585) = 119864(119889119884

119904+ 119892(119904 119883

119904 119884

119904 119885

119904)119889119904 |

119883119904 120585) We note that 119892(119904 119883

119904 119884

119904 119885

119904)119889119904 is a higher order

infinitesimal of 119885119904119889119861

119904when Δ tends to zero Under this

situation if 119892(119904 119883119904 119884

119904 119885

119904)119889119904 is ignored then

119898(119883119904 120585) ≐ 119864 (119889119884

119904| 119883

119904 120585) (10)

It implies that we can use ordinary nonparametric methodto estimate function 119898 For example we use the N-Wordinary nonparametric method to estimate119898(119883

119904 120585) valued

at (1199090 120585

0)

(1199090 120585

0)

=

sum119899minus1

119894=1sum119898

119895=1(119884

119894+1119895minus 119884

119894119895)119870

ℎ119883(119883

119894119895minus 119909

0)119870

ℎ120585(120585

119894119895minus 120585

0)

sum119899minus1

119894=1sum119898

119895=1119870ℎ119883

(119883119894119895

minus 1199090)119870

ℎ120585(120585

119894119895minus 120585

0)

(11)

where 119870ℎ119883

= 119870(sdotℎ119883)ℎ

119883and 119870

ℎ120585= 119870(sdotℎ

120585)ℎ

120585are regular

kernel functions with ℎ119883and ℎ

120585being the corresponding

bandwidths

25 Estimation for Generator 119892 As was shown in the non-parametric instrumental variables estimator of Hall andHorowitz [11] (hereinafter HH) we can adopt a nonpara-metric quasi-instrumental variables estimation to estimatethe generator 119892 So in the section we summarize the HHestimator of 119892 in the model

119864 [minus119889119884119905minus 119898 (119883

119905 120585) | 119883

119905 120585] = 119864 [119892 (119905 119883

119905 119884

119905 119885

119905) 119889119905 | 119883

119905 120585]

(12)

Since (1199090 120585

0) and 119885

2

1199090 119895are the consistent estimator of

119898(1199090 120585

0) and 119885

2

1199090 119895 respectively we use them instead of

119898(119883119904 120585) and 119885

119904in the above model and we get

119864 [minus119889119884119904minus (119883

119904 120585) | 119883

119904 120585]

= 119864 [119892 (119904 119883119904 119884

119904 119885

119904) 119889119905 | 119883

119904 120585]

(13)

Because 119885119904is function of 119883

119904and 119884

119904 for simplicity of

presentation we denote 119892(119904 119883119904 119884

119904 119885

119904) = 119892(119883

119904 119884

119904)Thus the

model becomes

119864 [minus119889119884119904minus (119883

119904 120585) | 119883

119904 120585] = 119864 [119892 (119883

119904 119884

119904) 119889119905 | 119883

119904 120585]

(14)

Let Y119894= ((119884

119894+Δminus 119884

119894) minus (119883

119894 120585))Δ X

119894= 119883

119894 Z

119894= 119884

119894

W = 120585 and U119894= 119881

119894radicΔ the model becomes

Y119894= 119892 (X

119894Z

119894) + U

119894 119864 (U

119894| X

119894W

119894) = 0 (15)

It is assumed that the support of (XZW) is containedin [0 1]

3 This assumption can always be satisfied by ifnecessary carrying outmonotone increasing transformationsofXZ andW For example one can replaceXZ andW by

4 Mathematical Problems in Engineering

Φ(X) Φ(Z) and Φ(W) where Φ is the normal distributionfunction We take (Y XZWU) to be a vector where Y

and U are scalars X and W are supported on [0 1] and Z

is supported on [0 1] The model is

Y119894= 119892 (X

119894Z

119894) + U

119894 119864 (U

119894| Z

119894W

119894) = 0 (16)

where (Y119894X

119894Z

119894W

119894U

119894) for 119894 ge 1 are independent and

identically distributed as (Y XZWU) Thus X and Z areendogenous and exogenous explanatory variables respec-tively Data (Y

119894X

119894Z

119894W

119894U

119894) for 1 le 119894 le 119899 are observed

Let119891XZW denote the density of (XZW) write119891Z for thedensity of Z and for each 119909

1 119909

2isin [0 1]

119901 and put

119905119911(119909

1 119909

2) = int119891XZW (119909

1 119911 119908) 119891XZW (119909

2 119911 119908) 119889119908 (17)

Define the operator 119879119911on 119871

2[0 1]

119901 by

(119879119911120595) (119909) = int 119905

119911(120585 119909) 120595 (120585) 119889120585 (18)

It may be proved that for each 119911 for which 119879minus1

119911exists

119892 (119909 119911)

= 119891Z (119911) 119864W|Z

times 119864 (Y | Z = 119911W) (119879minus1

119911119891XZW) (119909 119911W) | Z = 119911

(19)

where 119864W|Z denotes the expectation with respect to thedistribution of W conditional on Z In this formulation(119879

minus1

119911119891XZW)(119909 119911W) denoted the result of applying 119879minus1

119911to the

function 119891XZW(sdot 119911W) and evaluating the resulting functionat 119909

To construct an estimator of 119892(119909 119911) given ℎ gt 0 and 119909 =

119909(1) and 120585 = 120585

(1) let 119870ℎ(119909 120585) = 119870

ℎ(119909

(119895)

120585(119895)

) put 119870ℎ(119911 120585)

analogously for 119911 and 120585 let ℎ119909 ℎ

119911gt 0 and define

119891XZW (119909 119911 119908)

=1

119899ℎ2119909ℎ119911

119899

sum

119894=1

119870ℎ119909

(119909 minusX119894 119909)119870

ℎ119911(119911 minus Z

119894 119911)119870

ℎ119909(119908 minusW

119894 119908)

119891minus119894

XZW (119909 119911 119908)

=1

(119899 minus 1) ℎ2119909ℎ119911

sum

1le119895le119899119895 = 119894

119870ℎ119909

(119909 minusX119895 119909)

times 119870ℎ119911

(119911 minus Z119895 119911)119870

ℎ119911(119908 minusW

119895 119908)

119911(119909

1 119909

2) = int119891XZW (119909

1 119911 119908) 119891XZW (119909

2 119911 119908) 119889119908

(119911120595) (119909 119911 119908) = int

119911(120585 119909) 120595 (120585 119911 119908) 119889120585

(20)

where120595 is a function from1198773 to a real lineThen the estimator

of 119892(119909 119911) is

119892 (119909 119911) =1

119899

119899

sum

119894=1

(+

119911119891minus119894

XZW) (119909 119911W119894) 119884

119894119870ℎ119911

(119911 minus Z119894 119911) (21)

3 Asymptotic Results

In this section we study the asymptotic properties of ourproposed estimators All proofs are presented in Appendix

31 Asymptotic results of119885119904 To give the asymptotic results of

119885119904 we need the following conditions

(a) 1198831 119883

119899are 120588-mixing dependent namely the 120588-

mixing coefficients 120588(119897) satisfy 120588(119897) rarr 0 as 119897 rarr infinwhere

120588 (119897) = sup119864(119883119894+119897119883119894)minus119864(119883119894+119897)119864(119883119894) = 0

1003816100381610038161003816119864 (119883119894+119897119883119894) minus 119864 (119883

119894+119897) 119864 (119883

119894)1003816100381610038161003816

radicVar (119883119894+119897)Var (119883

119894)

(22)

with119883119894= 119883(119905

119894)

(b) |119885119894| le 119862 (a s) uniformly for 119894 = 1 119899 where 119862 is a

positive constant and 119885119894= 119885(119905

119894)

(c) The continuous kernel function 119870(sdot) is symmetricabout 0 with a support of interval [minus1 1] and

int

1

minus1

119870 (119906) 119889119906 = 1 1205902

119870= int

1

minus1

1199062

119870 (119906) 119889119906 = 0

int

1

minus1

|119906|119895

119870119896

(119906) 119889119906 lt infin for 119895 le 119896 = 1 2

(23)

Condition (a) is commonly used for weakly dependentprocess see for example Kolmogorov and Rozanov [12]Bradley and Bryc [13] Lu and Lin [14] and Su and Lin [6]Condition (b) is also reasonable because as is shown by (10)119885119905can be regarded as the deviation between the adjacent two

observations Condition (c) is standard for nonparametrickernel function

Theorem 1 Besides conditions (a) (b) and (c) let119883

1 119883

119899 be an observation sequence on a stationary

120588-mixing Markov process with the 120588-mixing coefficientssatisfying 120588(119897) = 120588

119897 for 0 lt 120588 lt 1 Furthermore 1198831 119883

119899

have a common and probability density 119901(119909) and for eachinterior point 119909

0in the support of 119901(sdot) 119901(119909

0) gt 0 1198852

(1199090) gt 0

the functions 119901(119909) and 119885(119909) have continuous two derivativesin neighborhood of 119909

0 As 119899 rarr infin such that 119899ℎ rarr infin

119899ℎ5

rarr 0 and 119899ℎΔ2

rarr 0 then

radic119899ℎ (1198852

(1199090) minus 119885

2

(1199090))

119889

997888rarr (01198854

(1199090) 119869

119870

119901 (1199090)

) (24)

where 119869119870= int

1

minus1

1198702

(119906)119889119906 lt infin

The asymptotic result in Theorem 1 is standard fornonparametric kernel estimator and here undersmoothing isused to eliminate asymptotic bias

32 Asymptotic results of 119892(119909119911) This section gives con-ditions under which the HH estimator of the generator

Mathematical Problems in Engineering 5

119892 is asymptotically distributed as 119873(0 119868) The followingadditional notations are used

Define U119894

= Y119894

minus 119892(X119894Z

119894) 119878

1198991(119909 119911) =

119899minus1

sum119899

119894=1U119894+

119891(minus119894)

XZW(119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911) and 119878

1198992(119909 119911) =

119899minus1

sum119899

119894=1119892(X

119894Z

119894)

+

119891(minus119894)

XZW(119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911) Then

119892(119909 119911) = 1198781198991(119909 119911) + 119878

1198992(119909 119911) Define 119879+

= (119879+ 119886119899119868)

minus1 Write

1198781198991

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894(119879

+

119891XZW) (119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911)

+ 119899minus1

119899

sum

119894=1

U119894(

+

119891(minus119894)

XZW minus 119879+

119891XZW)

times (119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911)

= 11987811989911

(119909 119911) + 11987811989912

(119909 119911)

(25)

Define 119881119899(119909 119911) = 119899

minus1 Var[U(119879+

119891XZW)(119909 119911W)] It followsfrom a triangular array version of the Lindeberg-Levy centrallimit theorem that 119878

11989911(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0 1) as 119899 rarr

infin Therefore [119892(119909 119911) minus 119892(119909 119911)]radic119881119899(119909 119911)rarr

119889

119873(0 1) if[119878

11989912(119909 119911) + 119878

1198992(119909 119911) minus 119892(119909 119911)]radic119881

119899(119909 119911) = 119900

119901(1)

Assumption 2 The data Y119894X

119894 Z

119894W

119894are independently and

identically distributed as (Y XXW) where (XZW) issupported on [0 1]

3 and 119864[Y minus 119892(XZ) | WZ] = 0

Assumption 3 The distribution of (XZW) has a density119891XZW with respect to Lebesgue measure Moreover 119891XZW is119903 times differentiable with respect to any combination of itsarguments where derivatives at the boundary of [0 1]3 aredefined as one sided derivativesThe derivatives are boundedin absolute value by 119862 In addition 119892 is 119903 times differentiableon [0 1]

2 with derivatives at 0 and 1 defined as one sidedThe derivatives of 119892 are bounded in absolute value by 119862 Inaddition 119864[Y 2

| XZW] le 119862 and 119864[Y 2

| XZW] le 119862 and119864[U2

| ZW] ge 119862119880for some finite constant 119862

119880

Assumption 4 The constants 120572 and 120573 satisfy 120572 gt 1 120573 gt 12and 120573 minus 12 le 120572 lt 2120573 Moreover 119887

119895le 119862119895

minus120573 119895minus120572 le 119862120582119895 and

suminfin

119896=1|119889

119911119895119896| le 119862119895

minus1205722 for all 119895 ge 1 In addition there are finitestrictly positive constants 119862

1205821and 119862

1205822 such that 119862

1205821le 120582

119895le

1198621205822119895minus120572 for all 119895 ge 1

Assumption 5 The tuning parameters 119886119899and ℎ satisfy 119886

119899≍

119899minus(120588120572)(2120573+120572) and ℎ ≍ 119899

minus1 where 119903 isin [1198601015840

2 119860

1015840

3]

Assumption 6 119870ℎdenotes a generalized kernel function with

the properties 119870ℎ(119906 119905) = 0 if 119906 gt 119905 or 119906 lt 119905 minus 1 for all

119905 isin [0 1]ℎminus(119895+1)

int119905minus1

119905119906119895

119870ℎ(119906 119905)119889119906 = 1 if 119895 = 0 else 0 if

1 le 119895 le 119903 minus 1 For each 120585 isin [0 1] 119870ℎ(ℎ 120585) is supported

on [(120585 minus 1)ℎ 120585ℎ] cap 120581 where 120581 is a compact interval notdepending on 120585 Moreover

supℎgt0120585isin[01]119906isin120581

119870ℎ(ℎ119906 120585) |lt infin (26)

Assumption 7 Consider 119864W[119879+

119891XZW(119909 119911W)]2

≍

119864W[119879+

119891XZW(sdot sdotW)]2 and 119864W[119879

+

119891XZW(sdot sdotW)]2

≍

int1

0

119879+

119891XZW(sdot sdotW)2

119889119908

Theorem 8 Let Assumptions 2ndash7 hold Then

119892 (119909 119911) minus 119892 (119909 119911)

radic119881119899(119909 119911)

997888rarr119889

119873(0 119868) (27)

holds except possibly on a set of (119909 119911) values whose Lebesgueis 0

Corollary 9 Let Assumptions 2ndash7 hold And if 119881119899(119909 119911) is

replaced with the consistent estimator

119899(119909 119911) = 119899

minus1

119899

sum

119894=1

U2

119894[

+

119891minus119894

119909119908(119911W

119894)119870

119902ℎ119911(119911 minus Z

119894 119911)]

2

(28)

where U119894= Y

119894minus 119892(X

119894Z

119894) This yields the Studentized statistic

[119892(119909 119911) minus 119892(119909 119911)]radic119899(119909 119911) Then

119892 (119909 119911) minus 119892 (119909 119911)

radic119899(119909 119911)

997888rarr119889

119873(0 119868) (29)

holds except possibly on a set of (119909 119911) values whose Lebesgueis 0

As was shown in the remark given in the previoussection even the conditional mean of error of the model isnonzero and the newly proposed estimation is consistentbecause of themixing dependency for details see the proof ofTheorem 8 Furthermore because of the terminal conditionthe asymptotic variance is larger than that without the use ofthe terminal condition

4 Simulation Studies

In this section we investigate the finite-sample behaviors bysimulation

Example 10 We consider a simple FBSDE as

119889119884119905= (

120583 minus 119903

120590119885119905+ 119903119884

119905)119889119905 + 119885

119905119889119861

119905

≜ (119887119884119905+ 119888119885

119905) + 119885

119905119889119861

119905 119884

119879= 120585

(30)

where119883119905is Geometric Brownian motion for modeling stock

price satisfying

119889119883119905= 120583119883

119905119889119905 + 120590119883

119905119889119861

119905 119883

0= 119909 (31)

while the riskless asset is the same as formula (31) 119889119875119905

=

1199031198750119889119905

Firstly let 120583 = 01 120590 = 001 Δ = 012 119899 = 300119879 = 366 and 119899

0= 119899

1= 10 Obviously119885

119905= 119899

1120590119883

119905We adopt

Epanechnikov kernel defined by119870(119906) = 34(1minus1199062

)119868(|119906| le 1)

6 Mathematical Problems in Engineering

012

01

008

006

004

002

0

Curve of ZEstimated curve of Z

0 5 10 15 20 25 30 35 40

(a)

Curve of gEstimated curve of g

0 5 10 15 20 25 30 35 40

14

12

1

08

06

04

02

0

(b)

Figure 1 The real lines are the true curves of 119885119905and function 119892(119905) respectively and the dashed ones are estimated curves for them in

Example 10

where 119868(sdot) is the indicator function For bandwidth selectionvarious data-driven techniques have been developed suchas cross-validation the plug-in method and the empiricalbias method However these useful tools require additionalcomputation intensiveness In our simulation we simplyapply the rule of thumb bandwidth selector For bandwidthselection bandwidth ℎ = std(119909)119899minus15 The values of thetuning parameters are 119886

119899= 005 120572 = 12 120573 = 1 Figure 1

presents the estimated curves for diffusion 119885119905and drift 119892 by

one simulation

Example 11 According to the theory ofmathematical financewe represent a European call option by the following FBSDEsmodel

119889119883119904= 119887119883

119904119889119904 + 120590119883

119904119889119882

119904

119889119884119904= [119903119884

119904+ (119887 minus 119903) 120590

minus1

119885119904] 119889119904 + 119885

119904119889119882

119904

1198830= 119909 119884

119879= (119883

119879minus 119870)

+

119904 isin [0 119879]

(32)

Here 1198831199040le119904le119879

and 1198841199040le119904le119879

are the price processes of thestock and the option respectively and119870 is the striking priceat the expiration date 119879 119883

1199040le119904le119879

follows the geometricBrownian motion as

119889119883119904= 119887119883

119904119889119904 + 120590119883

119904119889119882

119904

1198830= 119909 119904 isin [0 119879]

(33)

We use the Euler scheme to generate the price series ofthe stock as

119883119894+1

minus 119883119894= 119887119883

119894Δ + 120590119883

119894Δ12

120598119894 119894 = 0 119899 minus 1 (34)

where 120598119894119899minus1

119894=0is an iid series with standard normality

The price series by Black Scholes formula is part of thesolution of the FBSDEs above at discrete time points that is

119884119894= 119883

119894119873(119889

119894

+) minus 119890

minus119903(119899minus119894)Δ

119870119873(119889119894

minus) (35)

which together with

119885119894= 120590119883

119894119873(119889

119894

+) (36)

gives us data generating formulae where

119873(119910) =1

radic2120587int

119910

minusinfin

119890minus11990922

119889119909 (37)

is a cumulative normal function and

119889119894

plusmn=ln (119883

119894119870) + (119903 plusmn 120590

2

2) ((119899 minus 119894) Δ)

120590radic(119899 minus 119894) Δ (38)

We produce the true curve of the drift coefficient by

119892119894= minus119903119884

119894minus (119887 minus 119903) 120590

minus1

119885119894 (39)

We first apply formulas (21) and (11) to estimate 119892119894and

1198852

119894 respectively We adopt Epanechnikov kernel defined by

119870(119906) = 34(1 minus 1199062

)119868(|119906| le 1) where 119868(sdot) is the indicatorfunction For bandwidth selection we simply apply the ruleof thumb bandwidth selector

ℎ = constant times std (1198840 119884

119899minus1) 119899

minus15 (40)

to implement the estimationLet 119870 = 110 119883

0= 100 119887 = 01 120590 = 018 119903 = 008

119879 = 60 and Δ = 1100 The bandwidth parameters ℎ = 606

and ℎ = 067 are used for estimation of119892119904and119885

119904 respectively

The values of the tuning parameters are 119886119899= 005 120572 = 12

and 120573 = 1 To see the performance of our estimationmethodthe simulated and the estimated curves of the two coefficientsof the backward equation are displayed in Figures 2 and 3

Mathematical Problems in Engineering 7

0

0

10 20 30 40 50 60minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

minus100

Curve of gEstimated curve of g

Figure 2 The simulated curve and the estimated curves of 119892119904in

Example 11

00

10 20 30 40 50 60

Curve of ZEstimated curve of Z

800

700

600

500

400

300

200

100

Figure 3 The simulated curve and the estimated curves of 119885119904in

Example 11

Appendix

A Proofs

Proof of Theorem 1 Denote C = 1198831 119883

119899 By the

Taylor expansion and formula (8) we have

119864 (1198852

(1199090) | C)

=sum119899minus1

119894=1Δminus1

119870ℎ(119883

119894minus 119909

0) 119864 ((119884

119894+1minus 119884

119894)2

| C)

sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0)

=sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0) (119885

2

119894+ 119874 (Δ))

sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0)

=int119870

ℎ(119883

119894minus 119909

0) (119885

2

(119909)+119874 (Δ))119901 (119909) 119889119909 (1+ 119874119901(119899ℎ)

minus12

)

int119870ℎ(119883

119894minus 119909

0) 119901 (119909) 119889119909 (1+119874

119901(119899ℎ)

minus12

)

= ( (1198852

(1199090) + 119874 (Δ))

times (119901 (1199090) + (12) ℎ

2

119901(2)

(1199090) 120590

2

119870+ 119900 (ℎ

2

))

times (1 + 119874119901(119899ℎ)

minus12

) )

times ( (119901 (1199090) + (12) ℎ

2

119901(2)

(1199090) 120590

2

119870+ 119900 (ℎ

2

))

times (1 + 119874119901(119899ℎ)

minus12

) )

minus1

= 1198852

(1199090) +

119901(2)

(1199090)

2119901 (1199090)ℎ2

1198852

(1199090) 120590

2

119870+ 119900 (ℎ

2

) + 119874 (Δ)

(A1)

Furthermore

Var (1198852

(1199090) | C)

=1

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

times

119899minus1

sum

119894=1

Δminus2

1198702

ℎ(119883

119894minus 119909

0)Var ((119884

119894+1minus 119884

119894)2

| C)

+

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

Δminus2 cov (119870

ℎ(119883

119894minus 119909

0) (119884

119894+1minus 119884

119894)

119870ℎ(119883

119894+119896minus 119909

0) (119884

119894+119896+1minus 119884

119894+119896) | C)

(A2)

From the conditions of Markov process and 120588-mixing coeffi-cient1003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

Δminus2 cov ( 119870

ℎ(119883

119894minus 119909

0) (119884

119894+1minus 119884

119894)

119870ℎ(119883

119894+119896minus 119909

0) (119884

119894+119896+1minus 119884

119894+119896) )

1003816100381610038161003816100381610038161003816100381610038161003816

=1

(119899 minus 1)2

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

10038161003816100381610038161003816100381610038161003816119864 ((Δ)

minus2

(119884119894+1

minus 119884119894)2

(119884119894+119896+1

minus 119884119894+119896

)2

times (119870ℎ(119883

119894minus 119909

0) minus 119864 (119870

ℎ(119883

119894minus 119909

0)))

times(119870ℎ(119883

119894+119896minus119909

0)minus119864 (119870

ℎ(119883

119894+119896minus119909

0))))

10038161003816100381610038161003816100381610038161003816

=1

(119899 minus 1)2

10038161003816100381610038161003816100381610038161003816119864 (119885

2

1198941198852

119894+119897(119870

ℎ(119883

119894minus 119909

0) minus 119864 (119870

ℎ(119883

119894minus 119909

0)))

times (119870ℎ(119883

119894+119896minus 119909

0) minus 119864 (119870

ℎ(119883

119894+119896minus 119909

0))))

10038161003816100381610038161003816100381610038161003816

+ 119874 (Δ)

le119862

(119899 minus 1)2

ℎ

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

120588119896

= 119874(1

119899ℎ) = 119900 (1)

(A3)

8 Mathematical Problems in Engineering

Note that (119884119894+1

minus 119884119894)radicΔ = 119892(119905

119894 119884

119894 119885

119894)radicΔ + 119885

119894120578119894 where

119864(120578119894) = 0 Var(120578

119894) = 1 Thus Var((119884

119894+1minus119884

119894)radicΔ) = 119885

4

119894+119874(Δ)

and furthermore

Var (1198852

(1199090) | C)

=sum119899minus1

119894=1Δminus2

1198702

ℎ(119883

119894minus 119909

0)Var ((119884

119894+1minus 119884

119894)2

| C)

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

+ 119874119901(1)

=sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0) (119885

4

(1199090) + 119874 (radicΔ))

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

+ 119874119901(1)

=1198854

(1199090) 119869

119870+ 119874 (radicΔ)

119899ℎ119901 (1199090)

(1 + 119874119901(119899ℎ)

minus12

)

(A4)

To our interest both the conditional expectation and varianceare independent onC so the condition could be erased

From Lemma A1 of Politis and Romano [15] and therelation between the 120572-mixing condition and the 120588-mixingcondition (eg Theorem 111 of Lu and Lin [14]) we canensure that (119884

119894+1minus 119884

119894)2

119894 = 1 119899 minus 1 is a 120588-mixingdependent process and the mixing coefficient denoted by120588119884(119897) satisfies

infin

sum

119896=1

120588119884(2

119896

) le 119862

infin

sum

119896=1

120588 (2119896

) =

infin

sum

119896=1

1205882119896

lt infin (A5)

where119862 is a positive constant Finally we use the central limittheorems for 120588-mixing dependent process (eg Theorem401 of Lu and Lin [14]) to complete this proof

Proof of Theorem 8 Theorem 8 follows from proving that1198781198991(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0 1198682) and [119878

1198992(119909 119911) minus 119892(119909 119911)]

radic119881119899(119909 119911) = 119900

119901(1) except possibly if (119909 119911) belongs to a

set of Lebesgue measure 0 The first result is established inLemma A1 and the second is established in Lemma A2Throughout this Appendix ldquofor almost every (119909 119911)rdquo meansldquofor every (119909 119911) isin [0 1]

2 except possibly a set of Lebesguemeasure 0rdquo We make repeated use of the fact that if 1198641205952 =

119874(119899minus119904

) for some 119904 gt 0 then120595(119909 119911) = 119900119901(119899

minus119904

) for almost every(119909 119911)

Lemma A1 (asymptotic normality of 1198781198991(119909 119911)radic119881

119899(119909 119911))

Let Assumptions 2ndash7 hold Then 1198781198991(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0

1198682) for almost every (119909 119911)

Proof Define 11987811989911

(119909 119911) = 119899minus1

sum119899

119894=1U119894(119879

+

119891XZW)(119909 119911W119894)

1198601198992

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[119879

+

(119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

1198601198993

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[(

+

minus 119879+

) 119891XZW] (119909 119911W119894)

1198601198994

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[(

+

minus 119879+

) (119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

(A6)

Then 1198781198991(119909 119911) = 119878

11989911(119909 119911)+119860

1198992(119909 119911)+119860

1198993(119909 119911)+119860

1198994(119909 119911)

11987811989911

(119909 119911)radic119881119899(119909 119911)rarr

119889

119873(0 1198682) by a triangular array version

of the Lindeberg-Levy central limit theorem The proof ofthe triangular-array version of the theorem is identical to theproof of the ordinary Lindeberg-Levy theorem The lemmafollows if we can prove that 119860

119899119895(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for

119895 = 2 3 4 and almost every (119909 119911) isin [0 1]2

Assumption 7 and arguments like those leading to (62)of HH [11] show that

∬

1

0

119881119899(119909 119911) 119889119909 119889119911 ≍ 119899

minus[2120573+120572minus120588(120572+1)](2120573+120572)

(A7)

It follows from the Cauchy-Schwartz inequality 119864(119891(minus119894)

XZW minus

119891XZW) = 119874(ℎ1015840

) and Var(119891(minus119894)

XZW) = 119874[1(119899ℎ2

)] that

11986410038171003817100381710038171198601198992

10038171003817100381710038172

= 119874(1

1198992ℎ21198862119899

+ℎ2119903

1198991198862119899

) (A8)

Therefore it follows from Assumptions 5 and 7 that119860

1198992(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909 119911) Now

consider 1198601198993(119909 119911) Define the operator Δ = minus 119879 Then

1198601198993

(119909 119911) = minus ( + 119886119899119868) Δ119860

1198991(119909 119911) (A9)

Therefore the Cauchy-Schwartz inequality gives

11986410038171003817100381710038171198601198992

10038171003817100381710038172

le 11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

119864100381710038171003817100381711987811989911

10038171003817100381710038172

= 11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

∬

1

0

119881119899(119909 119911) 119889119909 119889119911

(A10)

HH show that

11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

= 119874(1

119899ℎ1198862119899

+ℎ2119903

1198862119899

) (A11)

Therefore it follows from Assumptions 5 and 7 that119860

1198993(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909 119911) Finally

some algebra shows that

1198601198994

(119909 119911) = minus( + 119886119899119868)

minus1

Δ1198601198992

(119909 119911) (A12)

Therefore 1198601198994(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909

119911) follows from (A11) and 1198601198992(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1)

Lemma A2 (asymptotic negligibility of 1198781198992(119909 119911) minus 119892(119909 119911))

Let Assumptions 2ndash7 hold Then 1198781198992(119909 119911) minus 119892(119909 119911)

radic119881119899(119909 119911) = 119900

119901(1) for almost every (119909 119911)

Mathematical Problems in Engineering 9

Proof Define

119863119899(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times (119891XZW minus 119891XZW) (119909 119911 119908) 119889120579 119889120578 119889119908

1198601198991

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894 119885

119894) (119879

+

119891XZW) (119909 119911W119894)

(A13)

Redefine

1198601198992

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

minus 119863119899(119909 119911)

1198601198993

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) 119891XZW] (119909 119911W119894) + 119863

119899(119909 119911)

1198601198994

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) (119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894)

(A14)

Then 1198781198992(119909 119911) = sum

4

119895=1119860

119899119895(119909 119911) Arguments identical to

those used to derive (62) and (63) of HH [11] show that119864119860

1198991minus 119892

2

= 119874[119899minus120588(21205731)(2120573+120572)] and

∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874119899minus[2120573+120572minus120588(120572+1)](2120573+120572)

(A15)

Therefore it follows from Assumptions 5 and 7 that

[1198641198601198991

(119909 119911) minus 119892 (119909 119911)]

radic119881119899(119909 119911)

= 119900 (1) (A16)

119881minus1

119899(119909 119911)∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874 (1) (A17)

for almost every (119909 119911)Now consider 119860

1198992(119909 119911) Define

119863119899119894(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times(119891(minus119894)

XZWminus119891XZW)(119909 119911 119908) 119889120579 119889120578 119889119908

11986011989921

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894) minus 119863

119899119894(119909 119911)

(A18)

and 11986011989922

(119909 119911) = 119899minus1

sum119899

119894=1[119863

119899119894(119909 119911) minus 119863

119899(119909 119911)] HH show

that 11986411986011989921

2

= 119874((ℎ2119903

1198991198862

119899) + (1119899

2

ℎ2

1198862

119899)) and 119864119860

119899222

=

119874(11198992

1198862

119899) Therefore it follows from Assumptions 5 and 7

that

1198601198992

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A19)

for almost every (119909 119911) Now consider 1198601198993(119909 119911) Write

1198601198993

(119909 119911) = 11986011989931

(119909 119911) + 11986011989932

(119909 119911) (A20)

where 11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ119892(119909 119911) + 119863119899(119909 119911) and

11986011989932

(119909 119911) = minus(+

+ 119886119899119868)

minus1

Δ(1198601198991

minus 119892)(119909 119911) It follows from(A11)-(A16) and (A20) that

11986011989932

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A21)

for almost every (119909 119911)To analyze 119860

11989931(119909 119911) define

1198611198991

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

1198611198992

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119909 119908) 119892 (119909 119909) 119889119909 119889119911 119889119908

1198611198993

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)

119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989911

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989912

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989921

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989922

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

(A22)

10 Mathematical Problems in Engineering

Define 120575 = ℎ2119903

+ (119899ℎ)minus1 HH show that

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

(11986111989921

+ 11986111989922

) (119909 119911)

(A23)

Define

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

11986111989921

(A24)

Then

119864100381710038171003817100381711986011989931

10038171003817100381710038172

le const [1198641003817100381710038171003817100381711986011989931

10038171003817100381710038171003817

2

+ 11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

]

(A25)

11986410038171003817100381710038171003817119860

11989931

10038171003817100381710038171003817

2

le const (1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

11986111989912

1003817100381710038171003817

4

+1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

(A26)

HH show that

1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817 = 119874(ℎ119903

119886119899

) (A27)

(1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

)12

= 119874(120575ℎ

2119903

119886119899

) (A28)

(1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

= 119874(1205752

1198862119899

) (A29)

See (611) (613) (614) and (615) of HH [11] Moreover

11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

= 119874(ℎ2119903minus1

1198991198862+(120572+1)120572

119899

+1

1198993ℎ51198864119899

+ℎ4119903

119899ℎ1198862119899

)

(A30)

See the arguments leading to (624) in HH [11] and theanalogous result for their equation (624) in HH [11] andthe analogous result for their quantity 119864119867

11989922 Combining

(A25)ndash(A30) with Assumptions 5 and 7 yields the result that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

=minus(119868 + 119879

+

Δ)minus1

119879+

11986111989912

radic119881119899(119909 119911)

+ 119900119901(1) (A31)

Now consider minus(119868 +119879+

Δ)minus1

119879+

11986111989912

Standard calculations forkernel estimators show that

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 119899minus1

119899

sum

119894=1

119891XZW (119909 119911W119894) 119892 (X

119894Z

119894) + 119874 (ℎ

119903

)

(A32)

Therefore

119879+

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 1198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

)

(A33)

119879+

11986111989912

(119909 119911) = 1198601198991

(119909 119911) minus 1198641198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

) (A34)

But

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911)

= 119879+

11986111989912

+ [(119868 + 119879+

Δ)minus1

minus 119868]119879+

11986111989912

= 119879+

11986111989912

+ ( + 119886119899119868)

minus1

Δ119879+

11986111989912

(A35)

Therefore it follows by combining Assumption 7 and equa-tions (A11) (A17) and (A34) that

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911) = 1198601198991

(119911) minus 1198641198601198991

(119909 119911) + 119903119899

(A36)

where 1198641199031198992

radic119881119899(119909 119911) = 119900(1) for almost every (119909 119911)

Combining this result with (A21) and (A31) gives

1198601198993

(119909 119911)

radic119881119899(119909 119911)

=minus [119860

1198991(119909 119911) minus 119864119860

1198991(119909 119911)]

radic119881119899(119909 119911)

+ 119900119901(1) (A37)

for almost every (119909 119911)Now consider 119860

1198994(119909 119911) HH show that

1198601198994

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ (1198601198992

minus 119879+

1198611198992) (119909 119911)

(A38)

Therefore it follows from (A19) and (A30) that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A39)

for almost every (119909 119911)Now combine (A19) (A37) and (A39) to obtain

1198781198992

(119909 119911)

radic119881119899(119909 119911)

=sum4

119895=1119860

119899119895(119909 119911)

radic119881119899(119909 119911)

=119864119860

1198991(119909 119911)

radic119881119899(119909 119911)

+ 119900119901(1)

(A40)

for almost every (119909 119911)The lemma follows by combining thisresult with (A16)

This completes the proof

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 no 1 pp 141ndash1831973

[3] F Antonelli ldquoBackward-forward stochastic differential equa-tionsrdquo The Annals of Applied Probability vol 3 no 3 pp 777ndash793 1993

[4] HWangW Li and XWang ldquoAsymptotic stabilization by statefeedback for a class of stochastic nonlinear systems with time-varying coefficientsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 258093 6 pages 2014

[5] W Zhang and G Li ldquoDiscrete-time indefinite stochastic linearquadratic optimal control with second moment constraintsrdquoMathematical Problems in Engineering vol 2014 Article ID278142 9 pages 2014

[6] Y Su and L Lin ldquoSemi-parametric estimation for forward-backward stochastic differential equationsrdquo Communications inStatistics Theory and Methods vol 38 no 11 pp 1759ndash17752009

[7] X Chen and L Lin ldquoNonparametric estimation for FBS-DEs models with applications in financerdquo Communications inStatisticsmdashTheory and Methods vol 39 no 14 pp 2492ndash25142010

[8] L Lin F Li and L X Zhu ldquoOn regressionwith variance built-inmean regression function a new financial modelrdquo Manuscript2009

[9] Q Zhang and L Lin ldquoTerminal-dependent statistical inferencesfor FBSDErdquo Stochastic Analysis and Applications vol 32 pp128ndash151 2014

[10] C Hsiao Analysis of Panel Data vol 36 of Econometric SocietyMonographs Cambridge University Press Cambridge UK 2ndedition 2003

[11] P Hall and J L Horowitz ldquoNonparametric methods for infer-ence in the presence of instrumental variablesrdquo The Annals ofStatistics vol 33 no 6 pp 2904ndash2929 2005

[12] A N Kolmogorov and U A Rozanov ldquoOn the strong mixingconditions of a stationary Gaussian processrdquo Theory of Proba-bility and Its Applications vol 2 pp 222ndash227 1960

[13] R C Bradley and W Bryc ldquoMultilinear forms and measures ofdependence between random variablesrdquo Journal of MultivariateAnalysis vol 16 no 3 pp 335ndash367 1985

[14] C R Lu and Z Y Lin Limit Theories for Mixing DependentVariables Science Press Beijing China 1997

[15] D N Politis and J P Romano ldquoA general resampling scheme fortriangular arrays of120572-mixing randomvariableswith applicationto the problem of spectral density estimationrdquo The Annals ofStatistics vol 20 no 4 pp 1985ndash2007 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Φ(X) Φ(Z) and Φ(W) where Φ is the normal distributionfunction We take (Y XZWU) to be a vector where Y

and U are scalars X and W are supported on [0 1] and Z

is supported on [0 1] The model is

Y119894= 119892 (X

119894Z

119894) + U

119894 119864 (U

119894| Z

119894W

119894) = 0 (16)

where (Y119894X

119894Z

119894W

119894U

119894) for 119894 ge 1 are independent and

identically distributed as (Y XZWU) Thus X and Z areendogenous and exogenous explanatory variables respec-tively Data (Y

119894X

119894Z

119894W

119894U

119894) for 1 le 119894 le 119899 are observed

Let119891XZW denote the density of (XZW) write119891Z for thedensity of Z and for each 119909

1 119909

2isin [0 1]

119901 and put

119905119911(119909

1 119909

2) = int119891XZW (119909

1 119911 119908) 119891XZW (119909

2 119911 119908) 119889119908 (17)

Define the operator 119879119911on 119871

2[0 1]

119901 by

(119879119911120595) (119909) = int 119905

119911(120585 119909) 120595 (120585) 119889120585 (18)

It may be proved that for each 119911 for which 119879minus1

119911exists

119892 (119909 119911)

= 119891Z (119911) 119864W|Z

times 119864 (Y | Z = 119911W) (119879minus1

119911119891XZW) (119909 119911W) | Z = 119911

(19)

where 119864W|Z denotes the expectation with respect to thedistribution of W conditional on Z In this formulation(119879

minus1

119911119891XZW)(119909 119911W) denoted the result of applying 119879minus1

119911to the

function 119891XZW(sdot 119911W) and evaluating the resulting functionat 119909

To construct an estimator of 119892(119909 119911) given ℎ gt 0 and 119909 =

119909(1) and 120585 = 120585

(1) let 119870ℎ(119909 120585) = 119870

ℎ(119909

(119895)

120585(119895)

) put 119870ℎ(119911 120585)

analogously for 119911 and 120585 let ℎ119909 ℎ

119911gt 0 and define

119891XZW (119909 119911 119908)

=1

119899ℎ2119909ℎ119911

119899

sum

119894=1

119870ℎ119909

(119909 minusX119894 119909)119870

ℎ119911(119911 minus Z

119894 119911)119870

ℎ119909(119908 minusW

119894 119908)

119891minus119894

XZW (119909 119911 119908)

=1

(119899 minus 1) ℎ2119909ℎ119911

sum

1le119895le119899119895 = 119894

119870ℎ119909

(119909 minusX119895 119909)

times 119870ℎ119911

(119911 minus Z119895 119911)119870

ℎ119911(119908 minusW

119895 119908)

119911(119909

1 119909

2) = int119891XZW (119909

1 119911 119908) 119891XZW (119909

2 119911 119908) 119889119908

(119911120595) (119909 119911 119908) = int

119911(120585 119909) 120595 (120585 119911 119908) 119889120585

(20)

where120595 is a function from1198773 to a real lineThen the estimator

of 119892(119909 119911) is

119892 (119909 119911) =1

119899

119899

sum

119894=1

(+

119911119891minus119894

XZW) (119909 119911W119894) 119884

119894119870ℎ119911

(119911 minus Z119894 119911) (21)

3 Asymptotic Results

In this section we study the asymptotic properties of ourproposed estimators All proofs are presented in Appendix

31 Asymptotic results of119885119904 To give the asymptotic results of

119885119904 we need the following conditions

(a) 1198831 119883

119899are 120588-mixing dependent namely the 120588-

mixing coefficients 120588(119897) satisfy 120588(119897) rarr 0 as 119897 rarr infinwhere

120588 (119897) = sup119864(119883119894+119897119883119894)minus119864(119883119894+119897)119864(119883119894) = 0

1003816100381610038161003816119864 (119883119894+119897119883119894) minus 119864 (119883

119894+119897) 119864 (119883

119894)1003816100381610038161003816

radicVar (119883119894+119897)Var (119883

119894)

(22)

with119883119894= 119883(119905

119894)

(b) |119885119894| le 119862 (a s) uniformly for 119894 = 1 119899 where 119862 is a

positive constant and 119885119894= 119885(119905

119894)

(c) The continuous kernel function 119870(sdot) is symmetricabout 0 with a support of interval [minus1 1] and

int

1

minus1

119870 (119906) 119889119906 = 1 1205902

119870= int

1

minus1

1199062

119870 (119906) 119889119906 = 0

int

1

minus1

|119906|119895

119870119896

(119906) 119889119906 lt infin for 119895 le 119896 = 1 2

(23)

Condition (a) is commonly used for weakly dependentprocess see for example Kolmogorov and Rozanov [12]Bradley and Bryc [13] Lu and Lin [14] and Su and Lin [6]Condition (b) is also reasonable because as is shown by (10)119885119905can be regarded as the deviation between the adjacent two

observations Condition (c) is standard for nonparametrickernel function

Theorem 1 Besides conditions (a) (b) and (c) let119883

1 119883

119899 be an observation sequence on a stationary

120588-mixing Markov process with the 120588-mixing coefficientssatisfying 120588(119897) = 120588

119897 for 0 lt 120588 lt 1 Furthermore 1198831 119883

119899

have a common and probability density 119901(119909) and for eachinterior point 119909

0in the support of 119901(sdot) 119901(119909

0) gt 0 1198852

(1199090) gt 0

the functions 119901(119909) and 119885(119909) have continuous two derivativesin neighborhood of 119909

0 As 119899 rarr infin such that 119899ℎ rarr infin

119899ℎ5

rarr 0 and 119899ℎΔ2

rarr 0 then

radic119899ℎ (1198852

(1199090) minus 119885

2

(1199090))

119889

997888rarr (01198854

(1199090) 119869

119870

119901 (1199090)

) (24)

where 119869119870= int

1

minus1

1198702

(119906)119889119906 lt infin

The asymptotic result in Theorem 1 is standard fornonparametric kernel estimator and here undersmoothing isused to eliminate asymptotic bias

32 Asymptotic results of 119892(119909119911) This section gives con-ditions under which the HH estimator of the generator

Mathematical Problems in Engineering 5

119892 is asymptotically distributed as 119873(0 119868) The followingadditional notations are used

Define U119894

= Y119894

minus 119892(X119894Z

119894) 119878

1198991(119909 119911) =

119899minus1

sum119899

119894=1U119894+

119891(minus119894)

XZW(119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911) and 119878

1198992(119909 119911) =

119899minus1

sum119899

119894=1119892(X

119894Z

119894)

+

119891(minus119894)

XZW(119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911) Then

119892(119909 119911) = 1198781198991(119909 119911) + 119878

1198992(119909 119911) Define 119879+

= (119879+ 119886119899119868)

minus1 Write

1198781198991

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894(119879

+

119891XZW) (119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911)

+ 119899minus1

119899

sum

119894=1

U119894(

+

119891(minus119894)

XZW minus 119879+

119891XZW)

times (119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911)

= 11987811989911

(119909 119911) + 11987811989912

(119909 119911)

(25)

Define 119881119899(119909 119911) = 119899

minus1 Var[U(119879+

119891XZW)(119909 119911W)] It followsfrom a triangular array version of the Lindeberg-Levy centrallimit theorem that 119878

11989911(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0 1) as 119899 rarr

infin Therefore [119892(119909 119911) minus 119892(119909 119911)]radic119881119899(119909 119911)rarr

119889

119873(0 1) if[119878

11989912(119909 119911) + 119878

1198992(119909 119911) minus 119892(119909 119911)]radic119881

119899(119909 119911) = 119900

119901(1)

Assumption 2 The data Y119894X

119894 Z

119894W

119894are independently and

identically distributed as (Y XXW) where (XZW) issupported on [0 1]

3 and 119864[Y minus 119892(XZ) | WZ] = 0

Assumption 3 The distribution of (XZW) has a density119891XZW with respect to Lebesgue measure Moreover 119891XZW is119903 times differentiable with respect to any combination of itsarguments where derivatives at the boundary of [0 1]3 aredefined as one sided derivativesThe derivatives are boundedin absolute value by 119862 In addition 119892 is 119903 times differentiableon [0 1]

2 with derivatives at 0 and 1 defined as one sidedThe derivatives of 119892 are bounded in absolute value by 119862 Inaddition 119864[Y 2

| XZW] le 119862 and 119864[Y 2

| XZW] le 119862 and119864[U2

| ZW] ge 119862119880for some finite constant 119862

119880

Assumption 4 The constants 120572 and 120573 satisfy 120572 gt 1 120573 gt 12and 120573 minus 12 le 120572 lt 2120573 Moreover 119887

119895le 119862119895

minus120573 119895minus120572 le 119862120582119895 and

suminfin

119896=1|119889

119911119895119896| le 119862119895

minus1205722 for all 119895 ge 1 In addition there are finitestrictly positive constants 119862

1205821and 119862

1205822 such that 119862

1205821le 120582

119895le

1198621205822119895minus120572 for all 119895 ge 1

Assumption 5 The tuning parameters 119886119899and ℎ satisfy 119886

119899≍

119899minus(120588120572)(2120573+120572) and ℎ ≍ 119899

minus1 where 119903 isin [1198601015840

2 119860

1015840

3]

Assumption 6 119870ℎdenotes a generalized kernel function with

the properties 119870ℎ(119906 119905) = 0 if 119906 gt 119905 or 119906 lt 119905 minus 1 for all

119905 isin [0 1]ℎminus(119895+1)

int119905minus1

119905119906119895

119870ℎ(119906 119905)119889119906 = 1 if 119895 = 0 else 0 if

1 le 119895 le 119903 minus 1 For each 120585 isin [0 1] 119870ℎ(ℎ 120585) is supported

on [(120585 minus 1)ℎ 120585ℎ] cap 120581 where 120581 is a compact interval notdepending on 120585 Moreover

supℎgt0120585isin[01]119906isin120581

119870ℎ(ℎ119906 120585) |lt infin (26)

Assumption 7 Consider 119864W[119879+

119891XZW(119909 119911W)]2

≍

119864W[119879+

119891XZW(sdot sdotW)]2 and 119864W[119879

+

119891XZW(sdot sdotW)]2

≍

int1

0

119879+

119891XZW(sdot sdotW)2

119889119908

Theorem 8 Let Assumptions 2ndash7 hold Then

119892 (119909 119911) minus 119892 (119909 119911)

radic119881119899(119909 119911)

997888rarr119889

119873(0 119868) (27)

holds except possibly on a set of (119909 119911) values whose Lebesgueis 0

Corollary 9 Let Assumptions 2ndash7 hold And if 119881119899(119909 119911) is

replaced with the consistent estimator

119899(119909 119911) = 119899

minus1

119899

sum

119894=1

U2

119894[

+

119891minus119894

119909119908(119911W

119894)119870

119902ℎ119911(119911 minus Z

119894 119911)]

2

(28)

where U119894= Y

119894minus 119892(X

119894Z

119894) This yields the Studentized statistic

[119892(119909 119911) minus 119892(119909 119911)]radic119899(119909 119911) Then

119892 (119909 119911) minus 119892 (119909 119911)

radic119899(119909 119911)

997888rarr119889

119873(0 119868) (29)

holds except possibly on a set of (119909 119911) values whose Lebesgueis 0

As was shown in the remark given in the previoussection even the conditional mean of error of the model isnonzero and the newly proposed estimation is consistentbecause of themixing dependency for details see the proof ofTheorem 8 Furthermore because of the terminal conditionthe asymptotic variance is larger than that without the use ofthe terminal condition

4 Simulation Studies

In this section we investigate the finite-sample behaviors bysimulation

Example 10 We consider a simple FBSDE as

119889119884119905= (

120583 minus 119903

120590119885119905+ 119903119884

119905)119889119905 + 119885

119905119889119861

119905

≜ (119887119884119905+ 119888119885

119905) + 119885

119905119889119861

119905 119884

119879= 120585

(30)

where119883119905is Geometric Brownian motion for modeling stock

price satisfying

119889119883119905= 120583119883

119905119889119905 + 120590119883

119905119889119861

119905 119883

0= 119909 (31)

while the riskless asset is the same as formula (31) 119889119875119905

=

1199031198750119889119905

Firstly let 120583 = 01 120590 = 001 Δ = 012 119899 = 300119879 = 366 and 119899

0= 119899

1= 10 Obviously119885

119905= 119899

1120590119883

119905We adopt

Epanechnikov kernel defined by119870(119906) = 34(1minus1199062

)119868(|119906| le 1)

6 Mathematical Problems in Engineering

012

01

008

006

004

002

0

Curve of ZEstimated curve of Z

0 5 10 15 20 25 30 35 40

(a)

Curve of gEstimated curve of g

0 5 10 15 20 25 30 35 40

14

12

1

08

06

04

02

0

(b)

Figure 1 The real lines are the true curves of 119885119905and function 119892(119905) respectively and the dashed ones are estimated curves for them in

Example 10

where 119868(sdot) is the indicator function For bandwidth selectionvarious data-driven techniques have been developed suchas cross-validation the plug-in method and the empiricalbias method However these useful tools require additionalcomputation intensiveness In our simulation we simplyapply the rule of thumb bandwidth selector For bandwidthselection bandwidth ℎ = std(119909)119899minus15 The values of thetuning parameters are 119886

119899= 005 120572 = 12 120573 = 1 Figure 1

presents the estimated curves for diffusion 119885119905and drift 119892 by

one simulation

Example 11 According to the theory ofmathematical financewe represent a European call option by the following FBSDEsmodel

119889119883119904= 119887119883

119904119889119904 + 120590119883

119904119889119882

119904

119889119884119904= [119903119884

119904+ (119887 minus 119903) 120590

minus1

119885119904] 119889119904 + 119885

119904119889119882

119904

1198830= 119909 119884

119879= (119883

119879minus 119870)

+

119904 isin [0 119879]

(32)

Here 1198831199040le119904le119879

and 1198841199040le119904le119879

are the price processes of thestock and the option respectively and119870 is the striking priceat the expiration date 119879 119883

1199040le119904le119879

follows the geometricBrownian motion as

119889119883119904= 119887119883

119904119889119904 + 120590119883

119904119889119882

119904

1198830= 119909 119904 isin [0 119879]

(33)

We use the Euler scheme to generate the price series ofthe stock as

119883119894+1

minus 119883119894= 119887119883

119894Δ + 120590119883

119894Δ12

120598119894 119894 = 0 119899 minus 1 (34)

where 120598119894119899minus1

119894=0is an iid series with standard normality

The price series by Black Scholes formula is part of thesolution of the FBSDEs above at discrete time points that is

119884119894= 119883

119894119873(119889

119894

+) minus 119890

minus119903(119899minus119894)Δ

119870119873(119889119894

minus) (35)

which together with

119885119894= 120590119883

119894119873(119889

119894

+) (36)

gives us data generating formulae where

119873(119910) =1

radic2120587int

119910

minusinfin

119890minus11990922

119889119909 (37)

is a cumulative normal function and

119889119894

plusmn=ln (119883

119894119870) + (119903 plusmn 120590

2

2) ((119899 minus 119894) Δ)

120590radic(119899 minus 119894) Δ (38)

We produce the true curve of the drift coefficient by

119892119894= minus119903119884

119894minus (119887 minus 119903) 120590

minus1

119885119894 (39)

We first apply formulas (21) and (11) to estimate 119892119894and

1198852

119894 respectively We adopt Epanechnikov kernel defined by

119870(119906) = 34(1 minus 1199062

)119868(|119906| le 1) where 119868(sdot) is the indicatorfunction For bandwidth selection we simply apply the ruleof thumb bandwidth selector

ℎ = constant times std (1198840 119884

119899minus1) 119899

minus15 (40)

to implement the estimationLet 119870 = 110 119883

0= 100 119887 = 01 120590 = 018 119903 = 008

119879 = 60 and Δ = 1100 The bandwidth parameters ℎ = 606

and ℎ = 067 are used for estimation of119892119904and119885

119904 respectively

The values of the tuning parameters are 119886119899= 005 120572 = 12

and 120573 = 1 To see the performance of our estimationmethodthe simulated and the estimated curves of the two coefficientsof the backward equation are displayed in Figures 2 and 3

Mathematical Problems in Engineering 7

0

0

10 20 30 40 50 60minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

minus100

Curve of gEstimated curve of g

Figure 2 The simulated curve and the estimated curves of 119892119904in

Example 11

00

10 20 30 40 50 60

Curve of ZEstimated curve of Z

800

700

600

500

400

300

200

100

Figure 3 The simulated curve and the estimated curves of 119885119904in

Example 11

Appendix

A Proofs

Proof of Theorem 1 Denote C = 1198831 119883

119899 By the

Taylor expansion and formula (8) we have

119864 (1198852

(1199090) | C)

=sum119899minus1

119894=1Δminus1

119870ℎ(119883

119894minus 119909

0) 119864 ((119884

119894+1minus 119884

119894)2

| C)

sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0)

=sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0) (119885

2

119894+ 119874 (Δ))

sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0)

=int119870

ℎ(119883

119894minus 119909

0) (119885

2

(119909)+119874 (Δ))119901 (119909) 119889119909 (1+ 119874119901(119899ℎ)

minus12

)

int119870ℎ(119883

119894minus 119909

0) 119901 (119909) 119889119909 (1+119874

119901(119899ℎ)

minus12

)

= ( (1198852

(1199090) + 119874 (Δ))

times (119901 (1199090) + (12) ℎ

2

119901(2)

(1199090) 120590

2

119870+ 119900 (ℎ

2

))

times (1 + 119874119901(119899ℎ)

minus12

) )

times ( (119901 (1199090) + (12) ℎ

2

119901(2)

(1199090) 120590

2

119870+ 119900 (ℎ

2

))

times (1 + 119874119901(119899ℎ)

minus12

) )

minus1

= 1198852

(1199090) +

119901(2)

(1199090)

2119901 (1199090)ℎ2

1198852

(1199090) 120590

2

119870+ 119900 (ℎ

2

) + 119874 (Δ)

(A1)

Furthermore

Var (1198852

(1199090) | C)

=1

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

times

119899minus1

sum

119894=1

Δminus2

1198702

ℎ(119883

119894minus 119909

0)Var ((119884

119894+1minus 119884

119894)2

| C)

+

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

Δminus2 cov (119870

ℎ(119883

119894minus 119909

0) (119884

119894+1minus 119884

119894)

119870ℎ(119883

119894+119896minus 119909

0) (119884

119894+119896+1minus 119884

119894+119896) | C)

(A2)

From the conditions of Markov process and 120588-mixing coeffi-cient1003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

Δminus2 cov ( 119870

ℎ(119883

119894minus 119909

0) (119884

119894+1minus 119884

119894)

119870ℎ(119883

119894+119896minus 119909

0) (119884

119894+119896+1minus 119884

119894+119896) )

1003816100381610038161003816100381610038161003816100381610038161003816

=1

(119899 minus 1)2

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

10038161003816100381610038161003816100381610038161003816119864 ((Δ)

minus2

(119884119894+1

minus 119884119894)2

(119884119894+119896+1

minus 119884119894+119896

)2

times (119870ℎ(119883

119894minus 119909

0) minus 119864 (119870

ℎ(119883

119894minus 119909

0)))

times(119870ℎ(119883

119894+119896minus119909

0)minus119864 (119870

ℎ(119883

119894+119896minus119909

0))))

10038161003816100381610038161003816100381610038161003816

=1

(119899 minus 1)2

10038161003816100381610038161003816100381610038161003816119864 (119885

2

1198941198852

119894+119897(119870

ℎ(119883

119894minus 119909

0) minus 119864 (119870

ℎ(119883

119894minus 119909

0)))

times (119870ℎ(119883

119894+119896minus 119909

0) minus 119864 (119870

ℎ(119883

119894+119896minus 119909

0))))

10038161003816100381610038161003816100381610038161003816

+ 119874 (Δ)

le119862

(119899 minus 1)2

ℎ

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

120588119896

= 119874(1

119899ℎ) = 119900 (1)

(A3)

8 Mathematical Problems in Engineering

Note that (119884119894+1

minus 119884119894)radicΔ = 119892(119905

119894 119884

119894 119885

119894)radicΔ + 119885

119894120578119894 where

119864(120578119894) = 0 Var(120578

119894) = 1 Thus Var((119884

119894+1minus119884

119894)radicΔ) = 119885

4

119894+119874(Δ)

and furthermore

Var (1198852

(1199090) | C)

=sum119899minus1

119894=1Δminus2

1198702

ℎ(119883

119894minus 119909

0)Var ((119884

119894+1minus 119884

119894)2

| C)

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

+ 119874119901(1)

=sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0) (119885

4

(1199090) + 119874 (radicΔ))

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

+ 119874119901(1)

=1198854

(1199090) 119869

119870+ 119874 (radicΔ)

119899ℎ119901 (1199090)

(1 + 119874119901(119899ℎ)

minus12

)

(A4)

To our interest both the conditional expectation and varianceare independent onC so the condition could be erased

From Lemma A1 of Politis and Romano [15] and therelation between the 120572-mixing condition and the 120588-mixingcondition (eg Theorem 111 of Lu and Lin [14]) we canensure that (119884

119894+1minus 119884

119894)2

119894 = 1 119899 minus 1 is a 120588-mixingdependent process and the mixing coefficient denoted by120588119884(119897) satisfies

infin

sum

119896=1

120588119884(2

119896

) le 119862

infin

sum

119896=1

120588 (2119896

) =

infin

sum

119896=1

1205882119896

lt infin (A5)

where119862 is a positive constant Finally we use the central limittheorems for 120588-mixing dependent process (eg Theorem401 of Lu and Lin [14]) to complete this proof

Proof of Theorem 8 Theorem 8 follows from proving that1198781198991(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0 1198682) and [119878

1198992(119909 119911) minus 119892(119909 119911)]

radic119881119899(119909 119911) = 119900

119901(1) except possibly if (119909 119911) belongs to a

set of Lebesgue measure 0 The first result is established inLemma A1 and the second is established in Lemma A2Throughout this Appendix ldquofor almost every (119909 119911)rdquo meansldquofor every (119909 119911) isin [0 1]

2 except possibly a set of Lebesguemeasure 0rdquo We make repeated use of the fact that if 1198641205952 =

119874(119899minus119904

) for some 119904 gt 0 then120595(119909 119911) = 119900119901(119899

minus119904

) for almost every(119909 119911)

Lemma A1 (asymptotic normality of 1198781198991(119909 119911)radic119881

119899(119909 119911))

Let Assumptions 2ndash7 hold Then 1198781198991(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0

1198682) for almost every (119909 119911)

Proof Define 11987811989911

(119909 119911) = 119899minus1

sum119899

119894=1U119894(119879

+

119891XZW)(119909 119911W119894)

1198601198992

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[119879

+

(119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

1198601198993

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[(

+

minus 119879+

) 119891XZW] (119909 119911W119894)

1198601198994

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[(

+

minus 119879+

) (119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

(A6)

Then 1198781198991(119909 119911) = 119878

11989911(119909 119911)+119860

1198992(119909 119911)+119860

1198993(119909 119911)+119860

1198994(119909 119911)

11987811989911

(119909 119911)radic119881119899(119909 119911)rarr

119889

119873(0 1198682) by a triangular array version

of the Lindeberg-Levy central limit theorem The proof ofthe triangular-array version of the theorem is identical to theproof of the ordinary Lindeberg-Levy theorem The lemmafollows if we can prove that 119860

119899119895(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for

119895 = 2 3 4 and almost every (119909 119911) isin [0 1]2

Assumption 7 and arguments like those leading to (62)of HH [11] show that

∬

1

0

119881119899(119909 119911) 119889119909 119889119911 ≍ 119899

minus[2120573+120572minus120588(120572+1)](2120573+120572)

(A7)

It follows from the Cauchy-Schwartz inequality 119864(119891(minus119894)

XZW minus

119891XZW) = 119874(ℎ1015840

) and Var(119891(minus119894)

XZW) = 119874[1(119899ℎ2

)] that

11986410038171003817100381710038171198601198992

10038171003817100381710038172

= 119874(1

1198992ℎ21198862119899

+ℎ2119903

1198991198862119899

) (A8)

Therefore it follows from Assumptions 5 and 7 that119860

1198992(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909 119911) Now

consider 1198601198993(119909 119911) Define the operator Δ = minus 119879 Then

1198601198993

(119909 119911) = minus ( + 119886119899119868) Δ119860

1198991(119909 119911) (A9)

Therefore the Cauchy-Schwartz inequality gives

11986410038171003817100381710038171198601198992

10038171003817100381710038172

le 11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

119864100381710038171003817100381711987811989911

10038171003817100381710038172

= 11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

∬

1

0

119881119899(119909 119911) 119889119909 119889119911

(A10)

HH show that

11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

= 119874(1

119899ℎ1198862119899

+ℎ2119903

1198862119899

) (A11)

Therefore it follows from Assumptions 5 and 7 that119860

1198993(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909 119911) Finally

some algebra shows that

1198601198994

(119909 119911) = minus( + 119886119899119868)

minus1

Δ1198601198992

(119909 119911) (A12)

Therefore 1198601198994(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909

119911) follows from (A11) and 1198601198992(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1)

Lemma A2 (asymptotic negligibility of 1198781198992(119909 119911) minus 119892(119909 119911))

Let Assumptions 2ndash7 hold Then 1198781198992(119909 119911) minus 119892(119909 119911)

radic119881119899(119909 119911) = 119900

119901(1) for almost every (119909 119911)

Mathematical Problems in Engineering 9

Proof Define

119863119899(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times (119891XZW minus 119891XZW) (119909 119911 119908) 119889120579 119889120578 119889119908

1198601198991

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894 119885

119894) (119879

+

119891XZW) (119909 119911W119894)

(A13)

Redefine

1198601198992

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

minus 119863119899(119909 119911)

1198601198993

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) 119891XZW] (119909 119911W119894) + 119863

119899(119909 119911)

1198601198994

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) (119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894)

(A14)

Then 1198781198992(119909 119911) = sum

4

119895=1119860

119899119895(119909 119911) Arguments identical to

those used to derive (62) and (63) of HH [11] show that119864119860

1198991minus 119892

2

= 119874[119899minus120588(21205731)(2120573+120572)] and

∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874119899minus[2120573+120572minus120588(120572+1)](2120573+120572)

(A15)

Therefore it follows from Assumptions 5 and 7 that

[1198641198601198991

(119909 119911) minus 119892 (119909 119911)]

radic119881119899(119909 119911)

= 119900 (1) (A16)

119881minus1

119899(119909 119911)∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874 (1) (A17)

for almost every (119909 119911)Now consider 119860

1198992(119909 119911) Define

119863119899119894(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times(119891(minus119894)

XZWminus119891XZW)(119909 119911 119908) 119889120579 119889120578 119889119908

11986011989921

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894) minus 119863

119899119894(119909 119911)

(A18)

and 11986011989922

(119909 119911) = 119899minus1

sum119899

119894=1[119863

119899119894(119909 119911) minus 119863

119899(119909 119911)] HH show

that 11986411986011989921

2

= 119874((ℎ2119903

1198991198862

119899) + (1119899

2

ℎ2

1198862

119899)) and 119864119860

119899222

=

119874(11198992

1198862

119899) Therefore it follows from Assumptions 5 and 7

that

1198601198992

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A19)

for almost every (119909 119911) Now consider 1198601198993(119909 119911) Write

1198601198993

(119909 119911) = 11986011989931

(119909 119911) + 11986011989932

(119909 119911) (A20)

where 11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ119892(119909 119911) + 119863119899(119909 119911) and

11986011989932

(119909 119911) = minus(+

+ 119886119899119868)

minus1

Δ(1198601198991

minus 119892)(119909 119911) It follows from(A11)-(A16) and (A20) that

11986011989932

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A21)

for almost every (119909 119911)To analyze 119860

11989931(119909 119911) define

1198611198991

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

1198611198992

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119909 119908) 119892 (119909 119909) 119889119909 119889119911 119889119908

1198611198993

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)

119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989911

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989912

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989921

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989922

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

(A22)

10 Mathematical Problems in Engineering

Define 120575 = ℎ2119903

+ (119899ℎ)minus1 HH show that

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

(11986111989921

+ 11986111989922

) (119909 119911)

(A23)

Define

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

11986111989921

(A24)

Then

119864100381710038171003817100381711986011989931

10038171003817100381710038172

le const [1198641003817100381710038171003817100381711986011989931

10038171003817100381710038171003817

2

+ 11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

]

(A25)

11986410038171003817100381710038171003817119860

11989931

10038171003817100381710038171003817

2

le const (1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

11986111989912

1003817100381710038171003817

4

+1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

(A26)

HH show that

1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817 = 119874(ℎ119903

119886119899

) (A27)

(1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

)12

= 119874(120575ℎ

2119903

119886119899

) (A28)

(1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

= 119874(1205752

1198862119899

) (A29)

See (611) (613) (614) and (615) of HH [11] Moreover

11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

= 119874(ℎ2119903minus1

1198991198862+(120572+1)120572

119899

+1

1198993ℎ51198864119899

+ℎ4119903

119899ℎ1198862119899

)

(A30)

See the arguments leading to (624) in HH [11] and theanalogous result for their equation (624) in HH [11] andthe analogous result for their quantity 119864119867

11989922 Combining

(A25)ndash(A30) with Assumptions 5 and 7 yields the result that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

=minus(119868 + 119879

+

Δ)minus1

119879+

11986111989912

radic119881119899(119909 119911)

+ 119900119901(1) (A31)

Now consider minus(119868 +119879+

Δ)minus1

119879+

11986111989912

Standard calculations forkernel estimators show that

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 119899minus1

119899

sum

119894=1

119891XZW (119909 119911W119894) 119892 (X

119894Z

119894) + 119874 (ℎ

119903

)

(A32)

Therefore

119879+

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 1198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

)

(A33)

119879+

11986111989912

(119909 119911) = 1198601198991

(119909 119911) minus 1198641198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

) (A34)

But

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911)

= 119879+

11986111989912

+ [(119868 + 119879+

Δ)minus1

minus 119868]119879+

11986111989912

= 119879+

11986111989912

+ ( + 119886119899119868)

minus1

Δ119879+

11986111989912

(A35)

Therefore it follows by combining Assumption 7 and equa-tions (A11) (A17) and (A34) that

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911) = 1198601198991

(119911) minus 1198641198601198991

(119909 119911) + 119903119899

(A36)

where 1198641199031198992

radic119881119899(119909 119911) = 119900(1) for almost every (119909 119911)

Combining this result with (A21) and (A31) gives

1198601198993

(119909 119911)

radic119881119899(119909 119911)

=minus [119860

1198991(119909 119911) minus 119864119860

1198991(119909 119911)]

radic119881119899(119909 119911)

+ 119900119901(1) (A37)

for almost every (119909 119911)Now consider 119860

1198994(119909 119911) HH show that

1198601198994

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ (1198601198992

minus 119879+

1198611198992) (119909 119911)

(A38)

Therefore it follows from (A19) and (A30) that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A39)

for almost every (119909 119911)Now combine (A19) (A37) and (A39) to obtain

1198781198992

(119909 119911)

radic119881119899(119909 119911)

=sum4

119895=1119860

119899119895(119909 119911)

radic119881119899(119909 119911)

=119864119860

1198991(119909 119911)

radic119881119899(119909 119911)

+ 119900119901(1)

(A40)

for almost every (119909 119911)The lemma follows by combining thisresult with (A16)

This completes the proof

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 no 1 pp 141ndash1831973

[3] F Antonelli ldquoBackward-forward stochastic differential equa-tionsrdquo The Annals of Applied Probability vol 3 no 3 pp 777ndash793 1993

[4] HWangW Li and XWang ldquoAsymptotic stabilization by statefeedback for a class of stochastic nonlinear systems with time-varying coefficientsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 258093 6 pages 2014

[5] W Zhang and G Li ldquoDiscrete-time indefinite stochastic linearquadratic optimal control with second moment constraintsrdquoMathematical Problems in Engineering vol 2014 Article ID278142 9 pages 2014

[6] Y Su and L Lin ldquoSemi-parametric estimation for forward-backward stochastic differential equationsrdquo Communications inStatistics Theory and Methods vol 38 no 11 pp 1759ndash17752009

[7] X Chen and L Lin ldquoNonparametric estimation for FBS-DEs models with applications in financerdquo Communications inStatisticsmdashTheory and Methods vol 39 no 14 pp 2492ndash25142010

[8] L Lin F Li and L X Zhu ldquoOn regressionwith variance built-inmean regression function a new financial modelrdquo Manuscript2009

[9] Q Zhang and L Lin ldquoTerminal-dependent statistical inferencesfor FBSDErdquo Stochastic Analysis and Applications vol 32 pp128ndash151 2014

[10] C Hsiao Analysis of Panel Data vol 36 of Econometric SocietyMonographs Cambridge University Press Cambridge UK 2ndedition 2003

[11] P Hall and J L Horowitz ldquoNonparametric methods for infer-ence in the presence of instrumental variablesrdquo The Annals ofStatistics vol 33 no 6 pp 2904ndash2929 2005

[12] A N Kolmogorov and U A Rozanov ldquoOn the strong mixingconditions of a stationary Gaussian processrdquo Theory of Proba-bility and Its Applications vol 2 pp 222ndash227 1960

[13] R C Bradley and W Bryc ldquoMultilinear forms and measures ofdependence between random variablesrdquo Journal of MultivariateAnalysis vol 16 no 3 pp 335ndash367 1985

[14] C R Lu and Z Y Lin Limit Theories for Mixing DependentVariables Science Press Beijing China 1997

[15] D N Politis and J P Romano ldquoA general resampling scheme fortriangular arrays of120572-mixing randomvariableswith applicationto the problem of spectral density estimationrdquo The Annals ofStatistics vol 20 no 4 pp 1985ndash2007 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

119892 is asymptotically distributed as 119873(0 119868) The followingadditional notations are used

Define U119894

= Y119894

minus 119892(X119894Z

119894) 119878

1198991(119909 119911) =

119899minus1

sum119899

119894=1U119894+

119891(minus119894)

XZW(119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911) and 119878

1198992(119909 119911) =

119899minus1

sum119899

119894=1119892(X

119894Z

119894)

+

119891(minus119894)

XZW(119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911) Then

119892(119909 119911) = 1198781198991(119909 119911) + 119878

1198992(119909 119911) Define 119879+

= (119879+ 119886119899119868)

minus1 Write

1198781198991

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894(119879

+

119891XZW) (119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911)

+ 119899minus1

119899

sum

119894=1

U119894(

+

119891(minus119894)

XZW minus 119879+

119891XZW)

times (119909 119911W119894)119870

119902ℎ119911(119911 minus Z

119894 119911)

= 11987811989911

(119909 119911) + 11987811989912

(119909 119911)

(25)

Define 119881119899(119909 119911) = 119899

minus1 Var[U(119879+

119891XZW)(119909 119911W)] It followsfrom a triangular array version of the Lindeberg-Levy centrallimit theorem that 119878

11989911(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0 1) as 119899 rarr

infin Therefore [119892(119909 119911) minus 119892(119909 119911)]radic119881119899(119909 119911)rarr

119889

119873(0 1) if[119878

11989912(119909 119911) + 119878

1198992(119909 119911) minus 119892(119909 119911)]radic119881

119899(119909 119911) = 119900

119901(1)

Assumption 2 The data Y119894X

119894 Z

119894W

119894are independently and

identically distributed as (Y XXW) where (XZW) issupported on [0 1]

3 and 119864[Y minus 119892(XZ) | WZ] = 0

Assumption 3 The distribution of (XZW) has a density119891XZW with respect to Lebesgue measure Moreover 119891XZW is119903 times differentiable with respect to any combination of itsarguments where derivatives at the boundary of [0 1]3 aredefined as one sided derivativesThe derivatives are boundedin absolute value by 119862 In addition 119892 is 119903 times differentiableon [0 1]

2 with derivatives at 0 and 1 defined as one sidedThe derivatives of 119892 are bounded in absolute value by 119862 Inaddition 119864[Y 2

| XZW] le 119862 and 119864[Y 2

| XZW] le 119862 and119864[U2

| ZW] ge 119862119880for some finite constant 119862

119880

Assumption 4 The constants 120572 and 120573 satisfy 120572 gt 1 120573 gt 12and 120573 minus 12 le 120572 lt 2120573 Moreover 119887

119895le 119862119895

minus120573 119895minus120572 le 119862120582119895 and

suminfin

119896=1|119889

119911119895119896| le 119862119895

minus1205722 for all 119895 ge 1 In addition there are finitestrictly positive constants 119862

1205821and 119862

1205822 such that 119862

1205821le 120582

119895le

1198621205822119895minus120572 for all 119895 ge 1

Assumption 5 The tuning parameters 119886119899and ℎ satisfy 119886

119899≍

119899minus(120588120572)(2120573+120572) and ℎ ≍ 119899

minus1 where 119903 isin [1198601015840

2 119860

1015840

3]

Assumption 6 119870ℎdenotes a generalized kernel function with

the properties 119870ℎ(119906 119905) = 0 if 119906 gt 119905 or 119906 lt 119905 minus 1 for all

119905 isin [0 1]ℎminus(119895+1)

int119905minus1

119905119906119895

119870ℎ(119906 119905)119889119906 = 1 if 119895 = 0 else 0 if

1 le 119895 le 119903 minus 1 For each 120585 isin [0 1] 119870ℎ(ℎ 120585) is supported

on [(120585 minus 1)ℎ 120585ℎ] cap 120581 where 120581 is a compact interval notdepending on 120585 Moreover

supℎgt0120585isin[01]119906isin120581

119870ℎ(ℎ119906 120585) |lt infin (26)

Assumption 7 Consider 119864W[119879+

119891XZW(119909 119911W)]2

≍

119864W[119879+

119891XZW(sdot sdotW)]2 and 119864W[119879

+

119891XZW(sdot sdotW)]2

≍

int1

0

119879+

119891XZW(sdot sdotW)2

119889119908

Theorem 8 Let Assumptions 2ndash7 hold Then

119892 (119909 119911) minus 119892 (119909 119911)

radic119881119899(119909 119911)

997888rarr119889

119873(0 119868) (27)

holds except possibly on a set of (119909 119911) values whose Lebesgueis 0

Corollary 9 Let Assumptions 2ndash7 hold And if 119881119899(119909 119911) is

replaced with the consistent estimator

119899(119909 119911) = 119899

minus1

119899

sum

119894=1

U2

119894[

+

119891minus119894

119909119908(119911W

119894)119870

119902ℎ119911(119911 minus Z

119894 119911)]

2

(28)

where U119894= Y

119894minus 119892(X

119894Z

119894) This yields the Studentized statistic

[119892(119909 119911) minus 119892(119909 119911)]radic119899(119909 119911) Then

119892 (119909 119911) minus 119892 (119909 119911)

radic119899(119909 119911)

997888rarr119889

119873(0 119868) (29)

holds except possibly on a set of (119909 119911) values whose Lebesgueis 0

As was shown in the remark given in the previoussection even the conditional mean of error of the model isnonzero and the newly proposed estimation is consistentbecause of themixing dependency for details see the proof ofTheorem 8 Furthermore because of the terminal conditionthe asymptotic variance is larger than that without the use ofthe terminal condition

4 Simulation Studies

In this section we investigate the finite-sample behaviors bysimulation

Example 10 We consider a simple FBSDE as

119889119884119905= (

120583 minus 119903

120590119885119905+ 119903119884

119905)119889119905 + 119885

119905119889119861

119905

≜ (119887119884119905+ 119888119885

119905) + 119885

119905119889119861

119905 119884

119879= 120585

(30)

where119883119905is Geometric Brownian motion for modeling stock

price satisfying

119889119883119905= 120583119883

119905119889119905 + 120590119883

119905119889119861

119905 119883

0= 119909 (31)

while the riskless asset is the same as formula (31) 119889119875119905

=

1199031198750119889119905

Firstly let 120583 = 01 120590 = 001 Δ = 012 119899 = 300119879 = 366 and 119899

0= 119899

1= 10 Obviously119885

119905= 119899

1120590119883

119905We adopt

Epanechnikov kernel defined by119870(119906) = 34(1minus1199062

)119868(|119906| le 1)

6 Mathematical Problems in Engineering

012

01

008

006

004

002

0

Curve of ZEstimated curve of Z

0 5 10 15 20 25 30 35 40

(a)

Curve of gEstimated curve of g

0 5 10 15 20 25 30 35 40

14

12

1

08

06

04

02

0

(b)

Figure 1 The real lines are the true curves of 119885119905and function 119892(119905) respectively and the dashed ones are estimated curves for them in

Example 10

where 119868(sdot) is the indicator function For bandwidth selectionvarious data-driven techniques have been developed suchas cross-validation the plug-in method and the empiricalbias method However these useful tools require additionalcomputation intensiveness In our simulation we simplyapply the rule of thumb bandwidth selector For bandwidthselection bandwidth ℎ = std(119909)119899minus15 The values of thetuning parameters are 119886

119899= 005 120572 = 12 120573 = 1 Figure 1

presents the estimated curves for diffusion 119885119905and drift 119892 by

one simulation

Example 11 According to the theory ofmathematical financewe represent a European call option by the following FBSDEsmodel

119889119883119904= 119887119883

119904119889119904 + 120590119883

119904119889119882

119904

119889119884119904= [119903119884

119904+ (119887 minus 119903) 120590

minus1

119885119904] 119889119904 + 119885

119904119889119882

119904

1198830= 119909 119884

119879= (119883

119879minus 119870)

+

119904 isin [0 119879]

(32)

Here 1198831199040le119904le119879

and 1198841199040le119904le119879

are the price processes of thestock and the option respectively and119870 is the striking priceat the expiration date 119879 119883

1199040le119904le119879

follows the geometricBrownian motion as

119889119883119904= 119887119883

119904119889119904 + 120590119883

119904119889119882

119904

1198830= 119909 119904 isin [0 119879]

(33)

We use the Euler scheme to generate the price series ofthe stock as

119883119894+1

minus 119883119894= 119887119883

119894Δ + 120590119883

119894Δ12

120598119894 119894 = 0 119899 minus 1 (34)

where 120598119894119899minus1

119894=0is an iid series with standard normality

The price series by Black Scholes formula is part of thesolution of the FBSDEs above at discrete time points that is

119884119894= 119883

119894119873(119889

119894

+) minus 119890

minus119903(119899minus119894)Δ

119870119873(119889119894

minus) (35)

which together with

119885119894= 120590119883

119894119873(119889

119894

+) (36)

gives us data generating formulae where

119873(119910) =1

radic2120587int

119910

minusinfin

119890minus11990922

119889119909 (37)

is a cumulative normal function and

119889119894

plusmn=ln (119883

119894119870) + (119903 plusmn 120590

2

2) ((119899 minus 119894) Δ)

120590radic(119899 minus 119894) Δ (38)

We produce the true curve of the drift coefficient by

119892119894= minus119903119884

119894minus (119887 minus 119903) 120590

minus1

119885119894 (39)

We first apply formulas (21) and (11) to estimate 119892119894and

1198852

119894 respectively We adopt Epanechnikov kernel defined by

119870(119906) = 34(1 minus 1199062

)119868(|119906| le 1) where 119868(sdot) is the indicatorfunction For bandwidth selection we simply apply the ruleof thumb bandwidth selector

ℎ = constant times std (1198840 119884

119899minus1) 119899

minus15 (40)

to implement the estimationLet 119870 = 110 119883

0= 100 119887 = 01 120590 = 018 119903 = 008

119879 = 60 and Δ = 1100 The bandwidth parameters ℎ = 606

and ℎ = 067 are used for estimation of119892119904and119885

119904 respectively

The values of the tuning parameters are 119886119899= 005 120572 = 12

and 120573 = 1 To see the performance of our estimationmethodthe simulated and the estimated curves of the two coefficientsof the backward equation are displayed in Figures 2 and 3

Mathematical Problems in Engineering 7

0

0

10 20 30 40 50 60minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

minus100

Curve of gEstimated curve of g

Figure 2 The simulated curve and the estimated curves of 119892119904in

Example 11

00

10 20 30 40 50 60

Curve of ZEstimated curve of Z

800

700

600

500

400

300

200

100

Figure 3 The simulated curve and the estimated curves of 119885119904in

Example 11

Appendix

A Proofs

Proof of Theorem 1 Denote C = 1198831 119883

119899 By the

Taylor expansion and formula (8) we have

119864 (1198852

(1199090) | C)

=sum119899minus1

119894=1Δminus1

119870ℎ(119883

119894minus 119909

0) 119864 ((119884

119894+1minus 119884

119894)2

| C)

sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0)

=sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0) (119885

2

119894+ 119874 (Δ))

sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0)

=int119870

ℎ(119883

119894minus 119909

0) (119885

2

(119909)+119874 (Δ))119901 (119909) 119889119909 (1+ 119874119901(119899ℎ)

minus12

)

int119870ℎ(119883

119894minus 119909

0) 119901 (119909) 119889119909 (1+119874

119901(119899ℎ)

minus12

)

= ( (1198852

(1199090) + 119874 (Δ))

times (119901 (1199090) + (12) ℎ

2

119901(2)

(1199090) 120590

2

119870+ 119900 (ℎ

2

))

times (1 + 119874119901(119899ℎ)

minus12

) )

times ( (119901 (1199090) + (12) ℎ

2

119901(2)

(1199090) 120590

2

119870+ 119900 (ℎ

2

))

times (1 + 119874119901(119899ℎ)

minus12

) )

minus1

= 1198852

(1199090) +

119901(2)

(1199090)

2119901 (1199090)ℎ2

1198852

(1199090) 120590

2

119870+ 119900 (ℎ

2

) + 119874 (Δ)

(A1)

Furthermore

Var (1198852

(1199090) | C)

=1

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

times

119899minus1

sum

119894=1

Δminus2

1198702

ℎ(119883

119894minus 119909

0)Var ((119884

119894+1minus 119884

119894)2

| C)

+

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

Δminus2 cov (119870

ℎ(119883

119894minus 119909

0) (119884

119894+1minus 119884

119894)

119870ℎ(119883

119894+119896minus 119909

0) (119884

119894+119896+1minus 119884

119894+119896) | C)

(A2)

From the conditions of Markov process and 120588-mixing coeffi-cient1003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

Δminus2 cov ( 119870

ℎ(119883

119894minus 119909

0) (119884

119894+1minus 119884

119894)

119870ℎ(119883

119894+119896minus 119909

0) (119884

119894+119896+1minus 119884

119894+119896) )

1003816100381610038161003816100381610038161003816100381610038161003816

=1

(119899 minus 1)2

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

10038161003816100381610038161003816100381610038161003816119864 ((Δ)

minus2

(119884119894+1

minus 119884119894)2

(119884119894+119896+1

minus 119884119894+119896

)2

times (119870ℎ(119883

119894minus 119909

0) minus 119864 (119870

ℎ(119883

119894minus 119909

0)))

times(119870ℎ(119883

119894+119896minus119909

0)minus119864 (119870

ℎ(119883

119894+119896minus119909

0))))

10038161003816100381610038161003816100381610038161003816

=1

(119899 minus 1)2

10038161003816100381610038161003816100381610038161003816119864 (119885

2

1198941198852

119894+119897(119870

ℎ(119883

119894minus 119909

0) minus 119864 (119870

ℎ(119883

119894minus 119909

0)))

times (119870ℎ(119883

119894+119896minus 119909

0) minus 119864 (119870

ℎ(119883

119894+119896minus 119909

0))))

10038161003816100381610038161003816100381610038161003816

+ 119874 (Δ)

le119862

(119899 minus 1)2

ℎ

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

120588119896

= 119874(1

119899ℎ) = 119900 (1)

(A3)

8 Mathematical Problems in Engineering

Note that (119884119894+1

minus 119884119894)radicΔ = 119892(119905

119894 119884

119894 119885

119894)radicΔ + 119885

119894120578119894 where

119864(120578119894) = 0 Var(120578

119894) = 1 Thus Var((119884

119894+1minus119884

119894)radicΔ) = 119885

4

119894+119874(Δ)

and furthermore

Var (1198852

(1199090) | C)

=sum119899minus1

119894=1Δminus2

1198702

ℎ(119883

119894minus 119909

0)Var ((119884

119894+1minus 119884

119894)2

| C)

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

+ 119874119901(1)

=sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0) (119885

4

(1199090) + 119874 (radicΔ))

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

+ 119874119901(1)

=1198854

(1199090) 119869

119870+ 119874 (radicΔ)

119899ℎ119901 (1199090)

(1 + 119874119901(119899ℎ)

minus12

)

(A4)

To our interest both the conditional expectation and varianceare independent onC so the condition could be erased

From Lemma A1 of Politis and Romano [15] and therelation between the 120572-mixing condition and the 120588-mixingcondition (eg Theorem 111 of Lu and Lin [14]) we canensure that (119884

119894+1minus 119884

119894)2

119894 = 1 119899 minus 1 is a 120588-mixingdependent process and the mixing coefficient denoted by120588119884(119897) satisfies

infin

sum

119896=1

120588119884(2

119896

) le 119862

infin

sum

119896=1

120588 (2119896

) =

infin

sum

119896=1

1205882119896

lt infin (A5)

where119862 is a positive constant Finally we use the central limittheorems for 120588-mixing dependent process (eg Theorem401 of Lu and Lin [14]) to complete this proof

Proof of Theorem 8 Theorem 8 follows from proving that1198781198991(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0 1198682) and [119878

1198992(119909 119911) minus 119892(119909 119911)]

radic119881119899(119909 119911) = 119900

119901(1) except possibly if (119909 119911) belongs to a

set of Lebesgue measure 0 The first result is established inLemma A1 and the second is established in Lemma A2Throughout this Appendix ldquofor almost every (119909 119911)rdquo meansldquofor every (119909 119911) isin [0 1]

2 except possibly a set of Lebesguemeasure 0rdquo We make repeated use of the fact that if 1198641205952 =

119874(119899minus119904

) for some 119904 gt 0 then120595(119909 119911) = 119900119901(119899

minus119904

) for almost every(119909 119911)

Lemma A1 (asymptotic normality of 1198781198991(119909 119911)radic119881

119899(119909 119911))

Let Assumptions 2ndash7 hold Then 1198781198991(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0

1198682) for almost every (119909 119911)

Proof Define 11987811989911

(119909 119911) = 119899minus1

sum119899

119894=1U119894(119879

+

119891XZW)(119909 119911W119894)

1198601198992

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[119879

+

(119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

1198601198993

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[(

+

minus 119879+

) 119891XZW] (119909 119911W119894)

1198601198994

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[(

+

minus 119879+

) (119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

(A6)

Then 1198781198991(119909 119911) = 119878

11989911(119909 119911)+119860

1198992(119909 119911)+119860

1198993(119909 119911)+119860

1198994(119909 119911)

11987811989911

(119909 119911)radic119881119899(119909 119911)rarr

119889

119873(0 1198682) by a triangular array version

of the Lindeberg-Levy central limit theorem The proof ofthe triangular-array version of the theorem is identical to theproof of the ordinary Lindeberg-Levy theorem The lemmafollows if we can prove that 119860

119899119895(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for

119895 = 2 3 4 and almost every (119909 119911) isin [0 1]2

Assumption 7 and arguments like those leading to (62)of HH [11] show that

∬

1

0

119881119899(119909 119911) 119889119909 119889119911 ≍ 119899

minus[2120573+120572minus120588(120572+1)](2120573+120572)

(A7)

It follows from the Cauchy-Schwartz inequality 119864(119891(minus119894)

XZW minus

119891XZW) = 119874(ℎ1015840

) and Var(119891(minus119894)

XZW) = 119874[1(119899ℎ2

)] that

11986410038171003817100381710038171198601198992

10038171003817100381710038172

= 119874(1

1198992ℎ21198862119899

+ℎ2119903

1198991198862119899

) (A8)

Therefore it follows from Assumptions 5 and 7 that119860

1198992(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909 119911) Now

consider 1198601198993(119909 119911) Define the operator Δ = minus 119879 Then

1198601198993

(119909 119911) = minus ( + 119886119899119868) Δ119860

1198991(119909 119911) (A9)

Therefore the Cauchy-Schwartz inequality gives

11986410038171003817100381710038171198601198992

10038171003817100381710038172

le 11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

119864100381710038171003817100381711987811989911

10038171003817100381710038172

= 11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

∬

1

0

119881119899(119909 119911) 119889119909 119889119911

(A10)

HH show that

11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

= 119874(1

119899ℎ1198862119899

+ℎ2119903

1198862119899

) (A11)

Therefore it follows from Assumptions 5 and 7 that119860

1198993(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909 119911) Finally

some algebra shows that

1198601198994

(119909 119911) = minus( + 119886119899119868)

minus1

Δ1198601198992

(119909 119911) (A12)

Therefore 1198601198994(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909

119911) follows from (A11) and 1198601198992(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1)

Lemma A2 (asymptotic negligibility of 1198781198992(119909 119911) minus 119892(119909 119911))

Let Assumptions 2ndash7 hold Then 1198781198992(119909 119911) minus 119892(119909 119911)

radic119881119899(119909 119911) = 119900

119901(1) for almost every (119909 119911)

Mathematical Problems in Engineering 9

Proof Define

119863119899(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times (119891XZW minus 119891XZW) (119909 119911 119908) 119889120579 119889120578 119889119908

1198601198991

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894 119885

119894) (119879

+

119891XZW) (119909 119911W119894)

(A13)

Redefine

1198601198992

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

minus 119863119899(119909 119911)

1198601198993

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) 119891XZW] (119909 119911W119894) + 119863

119899(119909 119911)

1198601198994

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) (119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894)

(A14)

Then 1198781198992(119909 119911) = sum

4

119895=1119860

119899119895(119909 119911) Arguments identical to

those used to derive (62) and (63) of HH [11] show that119864119860

1198991minus 119892

2

= 119874[119899minus120588(21205731)(2120573+120572)] and

∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874119899minus[2120573+120572minus120588(120572+1)](2120573+120572)

(A15)

Therefore it follows from Assumptions 5 and 7 that

[1198641198601198991

(119909 119911) minus 119892 (119909 119911)]

radic119881119899(119909 119911)

= 119900 (1) (A16)

119881minus1

119899(119909 119911)∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874 (1) (A17)

for almost every (119909 119911)Now consider 119860

1198992(119909 119911) Define

119863119899119894(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times(119891(minus119894)

XZWminus119891XZW)(119909 119911 119908) 119889120579 119889120578 119889119908

11986011989921

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894) minus 119863

119899119894(119909 119911)

(A18)

and 11986011989922

(119909 119911) = 119899minus1

sum119899

119894=1[119863

119899119894(119909 119911) minus 119863

119899(119909 119911)] HH show

that 11986411986011989921

2

= 119874((ℎ2119903

1198991198862

119899) + (1119899

2

ℎ2

1198862

119899)) and 119864119860

119899222

=

119874(11198992

1198862

119899) Therefore it follows from Assumptions 5 and 7

that

1198601198992

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A19)

for almost every (119909 119911) Now consider 1198601198993(119909 119911) Write

1198601198993

(119909 119911) = 11986011989931

(119909 119911) + 11986011989932

(119909 119911) (A20)

where 11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ119892(119909 119911) + 119863119899(119909 119911) and

11986011989932

(119909 119911) = minus(+

+ 119886119899119868)

minus1

Δ(1198601198991

minus 119892)(119909 119911) It follows from(A11)-(A16) and (A20) that

11986011989932

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A21)

for almost every (119909 119911)To analyze 119860

11989931(119909 119911) define

1198611198991

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

1198611198992

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119909 119908) 119892 (119909 119909) 119889119909 119889119911 119889119908

1198611198993

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)

119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989911

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989912

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989921

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989922

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

(A22)

10 Mathematical Problems in Engineering

Define 120575 = ℎ2119903

+ (119899ℎ)minus1 HH show that

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

(11986111989921

+ 11986111989922

) (119909 119911)

(A23)

Define

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

11986111989921

(A24)

Then

119864100381710038171003817100381711986011989931

10038171003817100381710038172

le const [1198641003817100381710038171003817100381711986011989931

10038171003817100381710038171003817

2

+ 11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

]

(A25)

11986410038171003817100381710038171003817119860

11989931

10038171003817100381710038171003817

2

le const (1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

11986111989912

1003817100381710038171003817

4

+1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

(A26)

HH show that

1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817 = 119874(ℎ119903

119886119899

) (A27)

(1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

)12

= 119874(120575ℎ

2119903

119886119899

) (A28)

(1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

= 119874(1205752

1198862119899

) (A29)

See (611) (613) (614) and (615) of HH [11] Moreover

11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

= 119874(ℎ2119903minus1

1198991198862+(120572+1)120572

119899

+1

1198993ℎ51198864119899

+ℎ4119903

119899ℎ1198862119899

)

(A30)

See the arguments leading to (624) in HH [11] and theanalogous result for their equation (624) in HH [11] andthe analogous result for their quantity 119864119867

11989922 Combining

(A25)ndash(A30) with Assumptions 5 and 7 yields the result that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

=minus(119868 + 119879

+

Δ)minus1

119879+

11986111989912

radic119881119899(119909 119911)

+ 119900119901(1) (A31)

Now consider minus(119868 +119879+

Δ)minus1

119879+

11986111989912

Standard calculations forkernel estimators show that

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 119899minus1

119899

sum

119894=1

119891XZW (119909 119911W119894) 119892 (X

119894Z

119894) + 119874 (ℎ

119903

)

(A32)

Therefore

119879+

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 1198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

)

(A33)

119879+

11986111989912

(119909 119911) = 1198601198991

(119909 119911) minus 1198641198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

) (A34)

But

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911)

= 119879+

11986111989912

+ [(119868 + 119879+

Δ)minus1

minus 119868]119879+

11986111989912

= 119879+

11986111989912

+ ( + 119886119899119868)

minus1

Δ119879+

11986111989912

(A35)

Therefore it follows by combining Assumption 7 and equa-tions (A11) (A17) and (A34) that

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911) = 1198601198991

(119911) minus 1198641198601198991

(119909 119911) + 119903119899

(A36)

where 1198641199031198992

radic119881119899(119909 119911) = 119900(1) for almost every (119909 119911)

Combining this result with (A21) and (A31) gives

1198601198993

(119909 119911)

radic119881119899(119909 119911)

=minus [119860

1198991(119909 119911) minus 119864119860

1198991(119909 119911)]

radic119881119899(119909 119911)

+ 119900119901(1) (A37)

for almost every (119909 119911)Now consider 119860

1198994(119909 119911) HH show that

1198601198994

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ (1198601198992

minus 119879+

1198611198992) (119909 119911)

(A38)

Therefore it follows from (A19) and (A30) that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A39)

for almost every (119909 119911)Now combine (A19) (A37) and (A39) to obtain

1198781198992

(119909 119911)

radic119881119899(119909 119911)

=sum4

119895=1119860

119899119895(119909 119911)

radic119881119899(119909 119911)

=119864119860

1198991(119909 119911)

radic119881119899(119909 119911)

+ 119900119901(1)

(A40)

for almost every (119909 119911)The lemma follows by combining thisresult with (A16)

This completes the proof

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 no 1 pp 141ndash1831973

[3] F Antonelli ldquoBackward-forward stochastic differential equa-tionsrdquo The Annals of Applied Probability vol 3 no 3 pp 777ndash793 1993

[4] HWangW Li and XWang ldquoAsymptotic stabilization by statefeedback for a class of stochastic nonlinear systems with time-varying coefficientsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 258093 6 pages 2014

[5] W Zhang and G Li ldquoDiscrete-time indefinite stochastic linearquadratic optimal control with second moment constraintsrdquoMathematical Problems in Engineering vol 2014 Article ID278142 9 pages 2014

[6] Y Su and L Lin ldquoSemi-parametric estimation for forward-backward stochastic differential equationsrdquo Communications inStatistics Theory and Methods vol 38 no 11 pp 1759ndash17752009

[7] X Chen and L Lin ldquoNonparametric estimation for FBS-DEs models with applications in financerdquo Communications inStatisticsmdashTheory and Methods vol 39 no 14 pp 2492ndash25142010

[8] L Lin F Li and L X Zhu ldquoOn regressionwith variance built-inmean regression function a new financial modelrdquo Manuscript2009

[9] Q Zhang and L Lin ldquoTerminal-dependent statistical inferencesfor FBSDErdquo Stochastic Analysis and Applications vol 32 pp128ndash151 2014

[10] C Hsiao Analysis of Panel Data vol 36 of Econometric SocietyMonographs Cambridge University Press Cambridge UK 2ndedition 2003

[11] P Hall and J L Horowitz ldquoNonparametric methods for infer-ence in the presence of instrumental variablesrdquo The Annals ofStatistics vol 33 no 6 pp 2904ndash2929 2005

[12] A N Kolmogorov and U A Rozanov ldquoOn the strong mixingconditions of a stationary Gaussian processrdquo Theory of Proba-bility and Its Applications vol 2 pp 222ndash227 1960

[13] R C Bradley and W Bryc ldquoMultilinear forms and measures ofdependence between random variablesrdquo Journal of MultivariateAnalysis vol 16 no 3 pp 335ndash367 1985

[14] C R Lu and Z Y Lin Limit Theories for Mixing DependentVariables Science Press Beijing China 1997

[15] D N Politis and J P Romano ldquoA general resampling scheme fortriangular arrays of120572-mixing randomvariableswith applicationto the problem of spectral density estimationrdquo The Annals ofStatistics vol 20 no 4 pp 1985ndash2007 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

012

01

008

006

004

002

0

Curve of ZEstimated curve of Z

0 5 10 15 20 25 30 35 40

(a)

Curve of gEstimated curve of g

0 5 10 15 20 25 30 35 40

14

12

1

08

06

04

02

0

(b)

Figure 1 The real lines are the true curves of 119885119905and function 119892(119905) respectively and the dashed ones are estimated curves for them in

Example 10

where 119868(sdot) is the indicator function For bandwidth selectionvarious data-driven techniques have been developed suchas cross-validation the plug-in method and the empiricalbias method However these useful tools require additionalcomputation intensiveness In our simulation we simplyapply the rule of thumb bandwidth selector For bandwidthselection bandwidth ℎ = std(119909)119899minus15 The values of thetuning parameters are 119886

119899= 005 120572 = 12 120573 = 1 Figure 1

presents the estimated curves for diffusion 119885119905and drift 119892 by

one simulation

Example 11 According to the theory ofmathematical financewe represent a European call option by the following FBSDEsmodel

119889119883119904= 119887119883

119904119889119904 + 120590119883

119904119889119882

119904

119889119884119904= [119903119884

119904+ (119887 minus 119903) 120590

minus1

119885119904] 119889119904 + 119885

119904119889119882

119904

1198830= 119909 119884

119879= (119883

119879minus 119870)

+

119904 isin [0 119879]

(32)

Here 1198831199040le119904le119879

and 1198841199040le119904le119879

are the price processes of thestock and the option respectively and119870 is the striking priceat the expiration date 119879 119883

1199040le119904le119879

follows the geometricBrownian motion as

119889119883119904= 119887119883

119904119889119904 + 120590119883

119904119889119882

119904

1198830= 119909 119904 isin [0 119879]

(33)

We use the Euler scheme to generate the price series ofthe stock as

119883119894+1

minus 119883119894= 119887119883

119894Δ + 120590119883

119894Δ12

120598119894 119894 = 0 119899 minus 1 (34)

where 120598119894119899minus1

119894=0is an iid series with standard normality

The price series by Black Scholes formula is part of thesolution of the FBSDEs above at discrete time points that is

119884119894= 119883

119894119873(119889

119894

+) minus 119890

minus119903(119899minus119894)Δ

119870119873(119889119894

minus) (35)

which together with

119885119894= 120590119883

119894119873(119889

119894

+) (36)

gives us data generating formulae where

119873(119910) =1

radic2120587int

119910

minusinfin

119890minus11990922

119889119909 (37)

is a cumulative normal function and

119889119894

plusmn=ln (119883

119894119870) + (119903 plusmn 120590

2

2) ((119899 minus 119894) Δ)

120590radic(119899 minus 119894) Δ (38)

We produce the true curve of the drift coefficient by

119892119894= minus119903119884

119894minus (119887 minus 119903) 120590

minus1

119885119894 (39)

We first apply formulas (21) and (11) to estimate 119892119894and

1198852

119894 respectively We adopt Epanechnikov kernel defined by

119870(119906) = 34(1 minus 1199062

)119868(|119906| le 1) where 119868(sdot) is the indicatorfunction For bandwidth selection we simply apply the ruleof thumb bandwidth selector

ℎ = constant times std (1198840 119884

119899minus1) 119899

minus15 (40)

to implement the estimationLet 119870 = 110 119883

0= 100 119887 = 01 120590 = 018 119903 = 008

119879 = 60 and Δ = 1100 The bandwidth parameters ℎ = 606

and ℎ = 067 are used for estimation of119892119904and119885

119904 respectively

The values of the tuning parameters are 119886119899= 005 120572 = 12

and 120573 = 1 To see the performance of our estimationmethodthe simulated and the estimated curves of the two coefficientsof the backward equation are displayed in Figures 2 and 3

Mathematical Problems in Engineering 7

0

0

10 20 30 40 50 60minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

minus100

Curve of gEstimated curve of g

Figure 2 The simulated curve and the estimated curves of 119892119904in

Example 11

00

10 20 30 40 50 60

Curve of ZEstimated curve of Z

800

700

600

500

400

300

200

100

Figure 3 The simulated curve and the estimated curves of 119885119904in

Example 11

Appendix

A Proofs

Proof of Theorem 1 Denote C = 1198831 119883

119899 By the

Taylor expansion and formula (8) we have

119864 (1198852

(1199090) | C)

=sum119899minus1

119894=1Δminus1

119870ℎ(119883

119894minus 119909

0) 119864 ((119884

119894+1minus 119884

119894)2

| C)

sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0)

=sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0) (119885

2

119894+ 119874 (Δ))

sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0)

=int119870

ℎ(119883

119894minus 119909

0) (119885

2

(119909)+119874 (Δ))119901 (119909) 119889119909 (1+ 119874119901(119899ℎ)

minus12

)

int119870ℎ(119883

119894minus 119909

0) 119901 (119909) 119889119909 (1+119874

119901(119899ℎ)

minus12

)

= ( (1198852

(1199090) + 119874 (Δ))

times (119901 (1199090) + (12) ℎ

2

119901(2)

(1199090) 120590

2

119870+ 119900 (ℎ

2

))

times (1 + 119874119901(119899ℎ)

minus12

) )

times ( (119901 (1199090) + (12) ℎ

2

119901(2)

(1199090) 120590

2

119870+ 119900 (ℎ

2

))

times (1 + 119874119901(119899ℎ)

minus12

) )

minus1

= 1198852

(1199090) +

119901(2)

(1199090)

2119901 (1199090)ℎ2

1198852

(1199090) 120590

2

119870+ 119900 (ℎ

2

) + 119874 (Δ)

(A1)

Furthermore

Var (1198852

(1199090) | C)

=1

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

times

119899minus1

sum

119894=1

Δminus2

1198702

ℎ(119883

119894minus 119909

0)Var ((119884

119894+1minus 119884

119894)2

| C)

+

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

Δminus2 cov (119870

ℎ(119883

119894minus 119909

0) (119884

119894+1minus 119884

119894)

119870ℎ(119883

119894+119896minus 119909

0) (119884

119894+119896+1minus 119884

119894+119896) | C)

(A2)

From the conditions of Markov process and 120588-mixing coeffi-cient1003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

Δminus2 cov ( 119870

ℎ(119883

119894minus 119909

0) (119884

119894+1minus 119884

119894)

119870ℎ(119883

119894+119896minus 119909

0) (119884

119894+119896+1minus 119884

119894+119896) )

1003816100381610038161003816100381610038161003816100381610038161003816

=1

(119899 minus 1)2

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

10038161003816100381610038161003816100381610038161003816119864 ((Δ)

minus2

(119884119894+1

minus 119884119894)2

(119884119894+119896+1

minus 119884119894+119896

)2

times (119870ℎ(119883

119894minus 119909

0) minus 119864 (119870

ℎ(119883

119894minus 119909

0)))

times(119870ℎ(119883

119894+119896minus119909

0)minus119864 (119870

ℎ(119883

119894+119896minus119909

0))))

10038161003816100381610038161003816100381610038161003816

=1

(119899 minus 1)2

10038161003816100381610038161003816100381610038161003816119864 (119885

2

1198941198852

119894+119897(119870

ℎ(119883

119894minus 119909

0) minus 119864 (119870

ℎ(119883

119894minus 119909

0)))

times (119870ℎ(119883

119894+119896minus 119909

0) minus 119864 (119870

ℎ(119883

119894+119896minus 119909

0))))

10038161003816100381610038161003816100381610038161003816

+ 119874 (Δ)

le119862

(119899 minus 1)2

ℎ

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

120588119896

= 119874(1

119899ℎ) = 119900 (1)

(A3)

8 Mathematical Problems in Engineering

Note that (119884119894+1

minus 119884119894)radicΔ = 119892(119905

119894 119884

119894 119885

119894)radicΔ + 119885

119894120578119894 where

119864(120578119894) = 0 Var(120578

119894) = 1 Thus Var((119884

119894+1minus119884

119894)radicΔ) = 119885

4

119894+119874(Δ)

and furthermore

Var (1198852

(1199090) | C)

=sum119899minus1

119894=1Δminus2

1198702

ℎ(119883

119894minus 119909

0)Var ((119884

119894+1minus 119884

119894)2

| C)

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

+ 119874119901(1)

=sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0) (119885

4

(1199090) + 119874 (radicΔ))

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

+ 119874119901(1)

=1198854

(1199090) 119869

119870+ 119874 (radicΔ)

119899ℎ119901 (1199090)

(1 + 119874119901(119899ℎ)

minus12

)

(A4)

To our interest both the conditional expectation and varianceare independent onC so the condition could be erased

From Lemma A1 of Politis and Romano [15] and therelation between the 120572-mixing condition and the 120588-mixingcondition (eg Theorem 111 of Lu and Lin [14]) we canensure that (119884

119894+1minus 119884

119894)2

119894 = 1 119899 minus 1 is a 120588-mixingdependent process and the mixing coefficient denoted by120588119884(119897) satisfies

infin

sum

119896=1

120588119884(2

119896

) le 119862

infin

sum

119896=1

120588 (2119896

) =

infin

sum

119896=1

1205882119896

lt infin (A5)

where119862 is a positive constant Finally we use the central limittheorems for 120588-mixing dependent process (eg Theorem401 of Lu and Lin [14]) to complete this proof

Proof of Theorem 8 Theorem 8 follows from proving that1198781198991(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0 1198682) and [119878

1198992(119909 119911) minus 119892(119909 119911)]

radic119881119899(119909 119911) = 119900

119901(1) except possibly if (119909 119911) belongs to a

set of Lebesgue measure 0 The first result is established inLemma A1 and the second is established in Lemma A2Throughout this Appendix ldquofor almost every (119909 119911)rdquo meansldquofor every (119909 119911) isin [0 1]

2 except possibly a set of Lebesguemeasure 0rdquo We make repeated use of the fact that if 1198641205952 =

119874(119899minus119904

) for some 119904 gt 0 then120595(119909 119911) = 119900119901(119899

minus119904

) for almost every(119909 119911)

Lemma A1 (asymptotic normality of 1198781198991(119909 119911)radic119881

119899(119909 119911))

Let Assumptions 2ndash7 hold Then 1198781198991(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0

1198682) for almost every (119909 119911)

Proof Define 11987811989911

(119909 119911) = 119899minus1

sum119899

119894=1U119894(119879

+

119891XZW)(119909 119911W119894)

1198601198992

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[119879

+

(119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

1198601198993

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[(

+

minus 119879+

) 119891XZW] (119909 119911W119894)

1198601198994

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[(

+

minus 119879+

) (119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

(A6)

Then 1198781198991(119909 119911) = 119878

11989911(119909 119911)+119860

1198992(119909 119911)+119860

1198993(119909 119911)+119860

1198994(119909 119911)

11987811989911

(119909 119911)radic119881119899(119909 119911)rarr

119889

119873(0 1198682) by a triangular array version

of the Lindeberg-Levy central limit theorem The proof ofthe triangular-array version of the theorem is identical to theproof of the ordinary Lindeberg-Levy theorem The lemmafollows if we can prove that 119860

119899119895(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for

119895 = 2 3 4 and almost every (119909 119911) isin [0 1]2

Assumption 7 and arguments like those leading to (62)of HH [11] show that

∬

1

0

119881119899(119909 119911) 119889119909 119889119911 ≍ 119899

minus[2120573+120572minus120588(120572+1)](2120573+120572)

(A7)

It follows from the Cauchy-Schwartz inequality 119864(119891(minus119894)

XZW minus

119891XZW) = 119874(ℎ1015840

) and Var(119891(minus119894)

XZW) = 119874[1(119899ℎ2

)] that

11986410038171003817100381710038171198601198992

10038171003817100381710038172

= 119874(1

1198992ℎ21198862119899

+ℎ2119903

1198991198862119899

) (A8)

Therefore it follows from Assumptions 5 and 7 that119860

1198992(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909 119911) Now

consider 1198601198993(119909 119911) Define the operator Δ = minus 119879 Then

1198601198993

(119909 119911) = minus ( + 119886119899119868) Δ119860

1198991(119909 119911) (A9)

Therefore the Cauchy-Schwartz inequality gives

11986410038171003817100381710038171198601198992

10038171003817100381710038172

le 11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

119864100381710038171003817100381711987811989911

10038171003817100381710038172

= 11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

∬

1

0

119881119899(119909 119911) 119889119909 119889119911

(A10)

HH show that

11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

= 119874(1

119899ℎ1198862119899

+ℎ2119903

1198862119899

) (A11)

Therefore it follows from Assumptions 5 and 7 that119860

1198993(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909 119911) Finally

some algebra shows that

1198601198994

(119909 119911) = minus( + 119886119899119868)

minus1

Δ1198601198992

(119909 119911) (A12)

Therefore 1198601198994(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909

119911) follows from (A11) and 1198601198992(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1)

Lemma A2 (asymptotic negligibility of 1198781198992(119909 119911) minus 119892(119909 119911))

Let Assumptions 2ndash7 hold Then 1198781198992(119909 119911) minus 119892(119909 119911)

radic119881119899(119909 119911) = 119900

119901(1) for almost every (119909 119911)

Mathematical Problems in Engineering 9

Proof Define

119863119899(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times (119891XZW minus 119891XZW) (119909 119911 119908) 119889120579 119889120578 119889119908

1198601198991

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894 119885

119894) (119879

+

119891XZW) (119909 119911W119894)

(A13)

Redefine

1198601198992

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

minus 119863119899(119909 119911)

1198601198993

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) 119891XZW] (119909 119911W119894) + 119863

119899(119909 119911)

1198601198994

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) (119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894)

(A14)

Then 1198781198992(119909 119911) = sum

4

119895=1119860

119899119895(119909 119911) Arguments identical to

those used to derive (62) and (63) of HH [11] show that119864119860

1198991minus 119892

2

= 119874[119899minus120588(21205731)(2120573+120572)] and

∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874119899minus[2120573+120572minus120588(120572+1)](2120573+120572)

(A15)

Therefore it follows from Assumptions 5 and 7 that

[1198641198601198991

(119909 119911) minus 119892 (119909 119911)]

radic119881119899(119909 119911)

= 119900 (1) (A16)

119881minus1

119899(119909 119911)∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874 (1) (A17)

for almost every (119909 119911)Now consider 119860

1198992(119909 119911) Define

119863119899119894(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times(119891(minus119894)

XZWminus119891XZW)(119909 119911 119908) 119889120579 119889120578 119889119908

11986011989921

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894) minus 119863

119899119894(119909 119911)

(A18)

and 11986011989922

(119909 119911) = 119899minus1

sum119899

119894=1[119863

119899119894(119909 119911) minus 119863

119899(119909 119911)] HH show

that 11986411986011989921

2

= 119874((ℎ2119903

1198991198862

119899) + (1119899

2

ℎ2

1198862

119899)) and 119864119860

119899222

=

119874(11198992

1198862

119899) Therefore it follows from Assumptions 5 and 7

that

1198601198992

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A19)

for almost every (119909 119911) Now consider 1198601198993(119909 119911) Write

1198601198993

(119909 119911) = 11986011989931

(119909 119911) + 11986011989932

(119909 119911) (A20)

where 11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ119892(119909 119911) + 119863119899(119909 119911) and

11986011989932

(119909 119911) = minus(+

+ 119886119899119868)

minus1

Δ(1198601198991

minus 119892)(119909 119911) It follows from(A11)-(A16) and (A20) that

11986011989932

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A21)

for almost every (119909 119911)To analyze 119860

11989931(119909 119911) define

1198611198991

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

1198611198992

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119909 119908) 119892 (119909 119909) 119889119909 119889119911 119889119908

1198611198993

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)

119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989911

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989912

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989921

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989922

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

(A22)

10 Mathematical Problems in Engineering

Define 120575 = ℎ2119903

+ (119899ℎ)minus1 HH show that

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

(11986111989921

+ 11986111989922

) (119909 119911)

(A23)

Define

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

11986111989921

(A24)

Then

119864100381710038171003817100381711986011989931

10038171003817100381710038172

le const [1198641003817100381710038171003817100381711986011989931

10038171003817100381710038171003817

2

+ 11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

]

(A25)

11986410038171003817100381710038171003817119860

11989931

10038171003817100381710038171003817

2

le const (1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

11986111989912

1003817100381710038171003817

4

+1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

(A26)

HH show that

1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817 = 119874(ℎ119903

119886119899

) (A27)

(1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

)12

= 119874(120575ℎ

2119903

119886119899

) (A28)

(1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

= 119874(1205752

1198862119899

) (A29)

See (611) (613) (614) and (615) of HH [11] Moreover

11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

= 119874(ℎ2119903minus1

1198991198862+(120572+1)120572

119899

+1

1198993ℎ51198864119899

+ℎ4119903

119899ℎ1198862119899

)

(A30)

See the arguments leading to (624) in HH [11] and theanalogous result for their equation (624) in HH [11] andthe analogous result for their quantity 119864119867

11989922 Combining

(A25)ndash(A30) with Assumptions 5 and 7 yields the result that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

=minus(119868 + 119879

+

Δ)minus1

119879+

11986111989912

radic119881119899(119909 119911)

+ 119900119901(1) (A31)

Now consider minus(119868 +119879+

Δ)minus1

119879+

11986111989912

Standard calculations forkernel estimators show that

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 119899minus1

119899

sum

119894=1

119891XZW (119909 119911W119894) 119892 (X

119894Z

119894) + 119874 (ℎ

119903

)

(A32)

Therefore

119879+

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 1198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

)

(A33)

119879+

11986111989912

(119909 119911) = 1198601198991

(119909 119911) minus 1198641198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

) (A34)

But

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911)

= 119879+

11986111989912

+ [(119868 + 119879+

Δ)minus1

minus 119868]119879+

11986111989912

= 119879+

11986111989912

+ ( + 119886119899119868)

minus1

Δ119879+

11986111989912

(A35)

Therefore it follows by combining Assumption 7 and equa-tions (A11) (A17) and (A34) that

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911) = 1198601198991

(119911) minus 1198641198601198991

(119909 119911) + 119903119899

(A36)

where 1198641199031198992

radic119881119899(119909 119911) = 119900(1) for almost every (119909 119911)

Combining this result with (A21) and (A31) gives

1198601198993

(119909 119911)

radic119881119899(119909 119911)

=minus [119860

1198991(119909 119911) minus 119864119860

1198991(119909 119911)]

radic119881119899(119909 119911)

+ 119900119901(1) (A37)

for almost every (119909 119911)Now consider 119860

1198994(119909 119911) HH show that

1198601198994

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ (1198601198992

minus 119879+

1198611198992) (119909 119911)

(A38)

Therefore it follows from (A19) and (A30) that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A39)

for almost every (119909 119911)Now combine (A19) (A37) and (A39) to obtain

1198781198992

(119909 119911)

radic119881119899(119909 119911)

=sum4

119895=1119860

119899119895(119909 119911)

radic119881119899(119909 119911)

=119864119860

1198991(119909 119911)

radic119881119899(119909 119911)

+ 119900119901(1)

(A40)

for almost every (119909 119911)The lemma follows by combining thisresult with (A16)

This completes the proof

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 no 1 pp 141ndash1831973

[3] F Antonelli ldquoBackward-forward stochastic differential equa-tionsrdquo The Annals of Applied Probability vol 3 no 3 pp 777ndash793 1993

[4] HWangW Li and XWang ldquoAsymptotic stabilization by statefeedback for a class of stochastic nonlinear systems with time-varying coefficientsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 258093 6 pages 2014

[5] W Zhang and G Li ldquoDiscrete-time indefinite stochastic linearquadratic optimal control with second moment constraintsrdquoMathematical Problems in Engineering vol 2014 Article ID278142 9 pages 2014

[6] Y Su and L Lin ldquoSemi-parametric estimation for forward-backward stochastic differential equationsrdquo Communications inStatistics Theory and Methods vol 38 no 11 pp 1759ndash17752009

[7] X Chen and L Lin ldquoNonparametric estimation for FBS-DEs models with applications in financerdquo Communications inStatisticsmdashTheory and Methods vol 39 no 14 pp 2492ndash25142010

[8] L Lin F Li and L X Zhu ldquoOn regressionwith variance built-inmean regression function a new financial modelrdquo Manuscript2009

[9] Q Zhang and L Lin ldquoTerminal-dependent statistical inferencesfor FBSDErdquo Stochastic Analysis and Applications vol 32 pp128ndash151 2014

[10] C Hsiao Analysis of Panel Data vol 36 of Econometric SocietyMonographs Cambridge University Press Cambridge UK 2ndedition 2003

[11] P Hall and J L Horowitz ldquoNonparametric methods for infer-ence in the presence of instrumental variablesrdquo The Annals ofStatistics vol 33 no 6 pp 2904ndash2929 2005

[12] A N Kolmogorov and U A Rozanov ldquoOn the strong mixingconditions of a stationary Gaussian processrdquo Theory of Proba-bility and Its Applications vol 2 pp 222ndash227 1960

[13] R C Bradley and W Bryc ldquoMultilinear forms and measures ofdependence between random variablesrdquo Journal of MultivariateAnalysis vol 16 no 3 pp 335ndash367 1985

[14] C R Lu and Z Y Lin Limit Theories for Mixing DependentVariables Science Press Beijing China 1997

[15] D N Politis and J P Romano ldquoA general resampling scheme fortriangular arrays of120572-mixing randomvariableswith applicationto the problem of spectral density estimationrdquo The Annals ofStatistics vol 20 no 4 pp 1985ndash2007 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0

0

10 20 30 40 50 60minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

minus100

Curve of gEstimated curve of g

Figure 2 The simulated curve and the estimated curves of 119892119904in

Example 11

00

10 20 30 40 50 60

Curve of ZEstimated curve of Z

800

700

600

500

400

300

200

100

Figure 3 The simulated curve and the estimated curves of 119885119904in

Example 11

Appendix

A Proofs

Proof of Theorem 1 Denote C = 1198831 119883

119899 By the

Taylor expansion and formula (8) we have

119864 (1198852

(1199090) | C)

=sum119899minus1

119894=1Δminus1

119870ℎ(119883

119894minus 119909

0) 119864 ((119884

119894+1minus 119884

119894)2

| C)

sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0)

=sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0) (119885

2

119894+ 119874 (Δ))

sum119899minus1

119894=1119870ℎ(119883

119894minus 119909

0)

=int119870

ℎ(119883

119894minus 119909

0) (119885

2

(119909)+119874 (Δ))119901 (119909) 119889119909 (1+ 119874119901(119899ℎ)

minus12

)

int119870ℎ(119883

119894minus 119909

0) 119901 (119909) 119889119909 (1+119874

119901(119899ℎ)

minus12

)

= ( (1198852

(1199090) + 119874 (Δ))

times (119901 (1199090) + (12) ℎ

2

119901(2)

(1199090) 120590

2

119870+ 119900 (ℎ

2

))

times (1 + 119874119901(119899ℎ)

minus12

) )

times ( (119901 (1199090) + (12) ℎ

2

119901(2)

(1199090) 120590

2

119870+ 119900 (ℎ

2

))

times (1 + 119874119901(119899ℎ)

minus12

) )

minus1

= 1198852

(1199090) +

119901(2)

(1199090)

2119901 (1199090)ℎ2

1198852

(1199090) 120590

2

119870+ 119900 (ℎ

2

) + 119874 (Δ)

(A1)

Furthermore

Var (1198852

(1199090) | C)

=1

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

times

119899minus1

sum

119894=1

Δminus2

1198702

ℎ(119883

119894minus 119909

0)Var ((119884

119894+1minus 119884

119894)2

| C)

+

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

Δminus2 cov (119870

ℎ(119883

119894minus 119909

0) (119884

119894+1minus 119884

119894)

119870ℎ(119883

119894+119896minus 119909

0) (119884

119894+119896+1minus 119884

119894+119896) | C)

(A2)

From the conditions of Markov process and 120588-mixing coeffi-cient1003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

Δminus2 cov ( 119870

ℎ(119883

119894minus 119909

0) (119884

119894+1minus 119884

119894)

119870ℎ(119883

119894+119896minus 119909

0) (119884

119894+119896+1minus 119884

119894+119896) )

1003816100381610038161003816100381610038161003816100381610038161003816

=1

(119899 minus 1)2

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

10038161003816100381610038161003816100381610038161003816119864 ((Δ)

minus2

(119884119894+1

minus 119884119894)2

(119884119894+119896+1

minus 119884119894+119896

)2

times (119870ℎ(119883

119894minus 119909

0) minus 119864 (119870

ℎ(119883

119894minus 119909

0)))

times(119870ℎ(119883

119894+119896minus119909

0)minus119864 (119870

ℎ(119883

119894+119896minus119909

0))))

10038161003816100381610038161003816100381610038161003816

=1

(119899 minus 1)2

10038161003816100381610038161003816100381610038161003816119864 (119885

2

1198941198852

119894+119897(119870

ℎ(119883

119894minus 119909

0) minus 119864 (119870

ℎ(119883

119894minus 119909

0)))

times (119870ℎ(119883

119894+119896minus 119909

0) minus 119864 (119870

ℎ(119883

119894+119896minus 119909

0))))

10038161003816100381610038161003816100381610038161003816

+ 119874 (Δ)

le119862

(119899 minus 1)2

ℎ

119899minus1

sum

119894=1

119899minus119894

sum

119896=1

120588119896

= 119874(1

119899ℎ) = 119900 (1)

(A3)

8 Mathematical Problems in Engineering

Note that (119884119894+1

minus 119884119894)radicΔ = 119892(119905

119894 119884

119894 119885

119894)radicΔ + 119885

119894120578119894 where

119864(120578119894) = 0 Var(120578

119894) = 1 Thus Var((119884

119894+1minus119884

119894)radicΔ) = 119885

4

119894+119874(Δ)

and furthermore

Var (1198852

(1199090) | C)

=sum119899minus1

119894=1Δminus2

1198702

ℎ(119883

119894minus 119909

0)Var ((119884

119894+1minus 119884

119894)2

| C)

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

+ 119874119901(1)

=sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0) (119885

4

(1199090) + 119874 (radicΔ))

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

+ 119874119901(1)

=1198854

(1199090) 119869

119870+ 119874 (radicΔ)

119899ℎ119901 (1199090)

(1 + 119874119901(119899ℎ)

minus12

)

(A4)

To our interest both the conditional expectation and varianceare independent onC so the condition could be erased

From Lemma A1 of Politis and Romano [15] and therelation between the 120572-mixing condition and the 120588-mixingcondition (eg Theorem 111 of Lu and Lin [14]) we canensure that (119884

119894+1minus 119884

119894)2

119894 = 1 119899 minus 1 is a 120588-mixingdependent process and the mixing coefficient denoted by120588119884(119897) satisfies

infin

sum

119896=1

120588119884(2

119896

) le 119862

infin

sum

119896=1

120588 (2119896

) =

infin

sum

119896=1

1205882119896

lt infin (A5)

where119862 is a positive constant Finally we use the central limittheorems for 120588-mixing dependent process (eg Theorem401 of Lu and Lin [14]) to complete this proof

Proof of Theorem 8 Theorem 8 follows from proving that1198781198991(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0 1198682) and [119878

1198992(119909 119911) minus 119892(119909 119911)]

radic119881119899(119909 119911) = 119900

119901(1) except possibly if (119909 119911) belongs to a

set of Lebesgue measure 0 The first result is established inLemma A1 and the second is established in Lemma A2Throughout this Appendix ldquofor almost every (119909 119911)rdquo meansldquofor every (119909 119911) isin [0 1]

2 except possibly a set of Lebesguemeasure 0rdquo We make repeated use of the fact that if 1198641205952 =

119874(119899minus119904

) for some 119904 gt 0 then120595(119909 119911) = 119900119901(119899

minus119904

) for almost every(119909 119911)

Lemma A1 (asymptotic normality of 1198781198991(119909 119911)radic119881

119899(119909 119911))

Let Assumptions 2ndash7 hold Then 1198781198991(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0

1198682) for almost every (119909 119911)

Proof Define 11987811989911

(119909 119911) = 119899minus1

sum119899

119894=1U119894(119879

+

119891XZW)(119909 119911W119894)

1198601198992

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[119879

+

(119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

1198601198993

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[(

+

minus 119879+

) 119891XZW] (119909 119911W119894)

1198601198994

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[(

+

minus 119879+

) (119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

(A6)

Then 1198781198991(119909 119911) = 119878

11989911(119909 119911)+119860

1198992(119909 119911)+119860

1198993(119909 119911)+119860

1198994(119909 119911)

11987811989911

(119909 119911)radic119881119899(119909 119911)rarr

119889

119873(0 1198682) by a triangular array version

of the Lindeberg-Levy central limit theorem The proof ofthe triangular-array version of the theorem is identical to theproof of the ordinary Lindeberg-Levy theorem The lemmafollows if we can prove that 119860

119899119895(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for

119895 = 2 3 4 and almost every (119909 119911) isin [0 1]2

Assumption 7 and arguments like those leading to (62)of HH [11] show that

∬

1

0

119881119899(119909 119911) 119889119909 119889119911 ≍ 119899

minus[2120573+120572minus120588(120572+1)](2120573+120572)

(A7)

It follows from the Cauchy-Schwartz inequality 119864(119891(minus119894)

XZW minus

119891XZW) = 119874(ℎ1015840

) and Var(119891(minus119894)

XZW) = 119874[1(119899ℎ2

)] that

11986410038171003817100381710038171198601198992

10038171003817100381710038172

= 119874(1

1198992ℎ21198862119899

+ℎ2119903

1198991198862119899

) (A8)

Therefore it follows from Assumptions 5 and 7 that119860

1198992(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909 119911) Now

consider 1198601198993(119909 119911) Define the operator Δ = minus 119879 Then

1198601198993

(119909 119911) = minus ( + 119886119899119868) Δ119860

1198991(119909 119911) (A9)

Therefore the Cauchy-Schwartz inequality gives

11986410038171003817100381710038171198601198992

10038171003817100381710038172

le 11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

119864100381710038171003817100381711987811989911

10038171003817100381710038172

= 11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

∬

1

0

119881119899(119909 119911) 119889119909 119889119911

(A10)

HH show that

11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

= 119874(1

119899ℎ1198862119899

+ℎ2119903

1198862119899

) (A11)

Therefore it follows from Assumptions 5 and 7 that119860

1198993(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909 119911) Finally

some algebra shows that

1198601198994

(119909 119911) = minus( + 119886119899119868)

minus1

Δ1198601198992

(119909 119911) (A12)

Therefore 1198601198994(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909

119911) follows from (A11) and 1198601198992(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1)

Lemma A2 (asymptotic negligibility of 1198781198992(119909 119911) minus 119892(119909 119911))

Let Assumptions 2ndash7 hold Then 1198781198992(119909 119911) minus 119892(119909 119911)

radic119881119899(119909 119911) = 119900

119901(1) for almost every (119909 119911)

Mathematical Problems in Engineering 9

Proof Define

119863119899(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times (119891XZW minus 119891XZW) (119909 119911 119908) 119889120579 119889120578 119889119908

1198601198991

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894 119885

119894) (119879

+

119891XZW) (119909 119911W119894)

(A13)

Redefine

1198601198992

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

minus 119863119899(119909 119911)

1198601198993

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) 119891XZW] (119909 119911W119894) + 119863

119899(119909 119911)

1198601198994

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) (119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894)

(A14)

Then 1198781198992(119909 119911) = sum

4

119895=1119860

119899119895(119909 119911) Arguments identical to

those used to derive (62) and (63) of HH [11] show that119864119860

1198991minus 119892

2

= 119874[119899minus120588(21205731)(2120573+120572)] and

∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874119899minus[2120573+120572minus120588(120572+1)](2120573+120572)

(A15)

Therefore it follows from Assumptions 5 and 7 that

[1198641198601198991

(119909 119911) minus 119892 (119909 119911)]

radic119881119899(119909 119911)

= 119900 (1) (A16)

119881minus1

119899(119909 119911)∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874 (1) (A17)

for almost every (119909 119911)Now consider 119860

1198992(119909 119911) Define

119863119899119894(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times(119891(minus119894)

XZWminus119891XZW)(119909 119911 119908) 119889120579 119889120578 119889119908

11986011989921

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894) minus 119863

119899119894(119909 119911)

(A18)

and 11986011989922

(119909 119911) = 119899minus1

sum119899

119894=1[119863

119899119894(119909 119911) minus 119863

119899(119909 119911)] HH show

that 11986411986011989921

2

= 119874((ℎ2119903

1198991198862

119899) + (1119899

2

ℎ2

1198862

119899)) and 119864119860

119899222

=

119874(11198992

1198862

119899) Therefore it follows from Assumptions 5 and 7

that

1198601198992

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A19)

for almost every (119909 119911) Now consider 1198601198993(119909 119911) Write

1198601198993

(119909 119911) = 11986011989931

(119909 119911) + 11986011989932

(119909 119911) (A20)

where 11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ119892(119909 119911) + 119863119899(119909 119911) and

11986011989932

(119909 119911) = minus(+

+ 119886119899119868)

minus1

Δ(1198601198991

minus 119892)(119909 119911) It follows from(A11)-(A16) and (A20) that

11986011989932

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A21)

for almost every (119909 119911)To analyze 119860

11989931(119909 119911) define

1198611198991

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

1198611198992

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119909 119908) 119892 (119909 119909) 119889119909 119889119911 119889119908

1198611198993

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)

119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989911

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989912

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989921

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989922

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

(A22)

10 Mathematical Problems in Engineering

Define 120575 = ℎ2119903

+ (119899ℎ)minus1 HH show that

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

(11986111989921

+ 11986111989922

) (119909 119911)

(A23)

Define

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

11986111989921

(A24)

Then

119864100381710038171003817100381711986011989931

10038171003817100381710038172

le const [1198641003817100381710038171003817100381711986011989931

10038171003817100381710038171003817

2

+ 11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

]

(A25)

11986410038171003817100381710038171003817119860

11989931

10038171003817100381710038171003817

2

le const (1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

11986111989912

1003817100381710038171003817

4

+1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

(A26)

HH show that

1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817 = 119874(ℎ119903

119886119899

) (A27)

(1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

)12

= 119874(120575ℎ

2119903

119886119899

) (A28)

(1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

= 119874(1205752

1198862119899

) (A29)

See (611) (613) (614) and (615) of HH [11] Moreover

11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

= 119874(ℎ2119903minus1

1198991198862+(120572+1)120572

119899

+1

1198993ℎ51198864119899

+ℎ4119903

119899ℎ1198862119899

)

(A30)

See the arguments leading to (624) in HH [11] and theanalogous result for their equation (624) in HH [11] andthe analogous result for their quantity 119864119867

11989922 Combining

(A25)ndash(A30) with Assumptions 5 and 7 yields the result that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

=minus(119868 + 119879

+

Δ)minus1

119879+

11986111989912

radic119881119899(119909 119911)

+ 119900119901(1) (A31)

Now consider minus(119868 +119879+

Δ)minus1

119879+

11986111989912

Standard calculations forkernel estimators show that

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 119899minus1

119899

sum

119894=1

119891XZW (119909 119911W119894) 119892 (X

119894Z

119894) + 119874 (ℎ

119903

)

(A32)

Therefore

119879+

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 1198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

)

(A33)

119879+

11986111989912

(119909 119911) = 1198601198991

(119909 119911) minus 1198641198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

) (A34)

But

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911)

= 119879+

11986111989912

+ [(119868 + 119879+

Δ)minus1

minus 119868]119879+

11986111989912

= 119879+

11986111989912

+ ( + 119886119899119868)

minus1

Δ119879+

11986111989912

(A35)

Therefore it follows by combining Assumption 7 and equa-tions (A11) (A17) and (A34) that

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911) = 1198601198991

(119911) minus 1198641198601198991

(119909 119911) + 119903119899

(A36)

where 1198641199031198992

radic119881119899(119909 119911) = 119900(1) for almost every (119909 119911)

Combining this result with (A21) and (A31) gives

1198601198993

(119909 119911)

radic119881119899(119909 119911)

=minus [119860

1198991(119909 119911) minus 119864119860

1198991(119909 119911)]

radic119881119899(119909 119911)

+ 119900119901(1) (A37)

for almost every (119909 119911)Now consider 119860

1198994(119909 119911) HH show that

1198601198994

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ (1198601198992

minus 119879+

1198611198992) (119909 119911)

(A38)

Therefore it follows from (A19) and (A30) that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A39)

for almost every (119909 119911)Now combine (A19) (A37) and (A39) to obtain

1198781198992

(119909 119911)

radic119881119899(119909 119911)

=sum4

119895=1119860

119899119895(119909 119911)

radic119881119899(119909 119911)

=119864119860

1198991(119909 119911)

radic119881119899(119909 119911)

+ 119900119901(1)

(A40)

for almost every (119909 119911)The lemma follows by combining thisresult with (A16)

This completes the proof

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 no 1 pp 141ndash1831973

[3] F Antonelli ldquoBackward-forward stochastic differential equa-tionsrdquo The Annals of Applied Probability vol 3 no 3 pp 777ndash793 1993

[4] HWangW Li and XWang ldquoAsymptotic stabilization by statefeedback for a class of stochastic nonlinear systems with time-varying coefficientsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 258093 6 pages 2014

[5] W Zhang and G Li ldquoDiscrete-time indefinite stochastic linearquadratic optimal control with second moment constraintsrdquoMathematical Problems in Engineering vol 2014 Article ID278142 9 pages 2014

[6] Y Su and L Lin ldquoSemi-parametric estimation for forward-backward stochastic differential equationsrdquo Communications inStatistics Theory and Methods vol 38 no 11 pp 1759ndash17752009

[7] X Chen and L Lin ldquoNonparametric estimation for FBS-DEs models with applications in financerdquo Communications inStatisticsmdashTheory and Methods vol 39 no 14 pp 2492ndash25142010

[8] L Lin F Li and L X Zhu ldquoOn regressionwith variance built-inmean regression function a new financial modelrdquo Manuscript2009

[9] Q Zhang and L Lin ldquoTerminal-dependent statistical inferencesfor FBSDErdquo Stochastic Analysis and Applications vol 32 pp128ndash151 2014

[10] C Hsiao Analysis of Panel Data vol 36 of Econometric SocietyMonographs Cambridge University Press Cambridge UK 2ndedition 2003

[11] P Hall and J L Horowitz ldquoNonparametric methods for infer-ence in the presence of instrumental variablesrdquo The Annals ofStatistics vol 33 no 6 pp 2904ndash2929 2005

[12] A N Kolmogorov and U A Rozanov ldquoOn the strong mixingconditions of a stationary Gaussian processrdquo Theory of Proba-bility and Its Applications vol 2 pp 222ndash227 1960

[13] R C Bradley and W Bryc ldquoMultilinear forms and measures ofdependence between random variablesrdquo Journal of MultivariateAnalysis vol 16 no 3 pp 335ndash367 1985

[14] C R Lu and Z Y Lin Limit Theories for Mixing DependentVariables Science Press Beijing China 1997

[15] D N Politis and J P Romano ldquoA general resampling scheme fortriangular arrays of120572-mixing randomvariableswith applicationto the problem of spectral density estimationrdquo The Annals ofStatistics vol 20 no 4 pp 1985ndash2007 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Note that (119884119894+1

minus 119884119894)radicΔ = 119892(119905

119894 119884

119894 119885

119894)radicΔ + 119885

119894120578119894 where

119864(120578119894) = 0 Var(120578

119894) = 1 Thus Var((119884

119894+1minus119884

119894)radicΔ) = 119885

4

119894+119874(Δ)

and furthermore

Var (1198852

(1199090) | C)

=sum119899minus1

119894=1Δminus2

1198702

ℎ(119883

119894minus 119909

0)Var ((119884

119894+1minus 119884

119894)2

| C)

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

+ 119874119901(1)

=sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0) (119885

4

(1199090) + 119874 (radicΔ))

sum119899minus1

119894=11198702

ℎ(119883

119894minus 119909

0)

+ 119874119901(1)

=1198854

(1199090) 119869

119870+ 119874 (radicΔ)

119899ℎ119901 (1199090)

(1 + 119874119901(119899ℎ)

minus12

)

(A4)

To our interest both the conditional expectation and varianceare independent onC so the condition could be erased

From Lemma A1 of Politis and Romano [15] and therelation between the 120572-mixing condition and the 120588-mixingcondition (eg Theorem 111 of Lu and Lin [14]) we canensure that (119884

119894+1minus 119884

119894)2

119894 = 1 119899 minus 1 is a 120588-mixingdependent process and the mixing coefficient denoted by120588119884(119897) satisfies

infin

sum

119896=1

120588119884(2

119896

) le 119862

infin

sum

119896=1

120588 (2119896

) =

infin

sum

119896=1

1205882119896

lt infin (A5)

where119862 is a positive constant Finally we use the central limittheorems for 120588-mixing dependent process (eg Theorem401 of Lu and Lin [14]) to complete this proof

Proof of Theorem 8 Theorem 8 follows from proving that1198781198991(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0 1198682) and [119878

1198992(119909 119911) minus 119892(119909 119911)]

radic119881119899(119909 119911) = 119900

119901(1) except possibly if (119909 119911) belongs to a

set of Lebesgue measure 0 The first result is established inLemma A1 and the second is established in Lemma A2Throughout this Appendix ldquofor almost every (119909 119911)rdquo meansldquofor every (119909 119911) isin [0 1]

2 except possibly a set of Lebesguemeasure 0rdquo We make repeated use of the fact that if 1198641205952 =

119874(119899minus119904

) for some 119904 gt 0 then120595(119909 119911) = 119900119901(119899

minus119904

) for almost every(119909 119911)

Lemma A1 (asymptotic normality of 1198781198991(119909 119911)radic119881

119899(119909 119911))

Let Assumptions 2ndash7 hold Then 1198781198991(119909 119911)radic119881

119899(119909 119911)rarr

119889

119873(0

1198682) for almost every (119909 119911)

Proof Define 11987811989911

(119909 119911) = 119899minus1

sum119899

119894=1U119894(119879

+

119891XZW)(119909 119911W119894)

1198601198992

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[119879

+

(119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

1198601198993

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[(

+

minus 119879+

) 119891XZW] (119909 119911W119894)

1198601198994

(119909 119911)

= 119899minus1

119899

sum

119894=1

U119894[(

+

minus 119879+

) (119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

(A6)

Then 1198781198991(119909 119911) = 119878

11989911(119909 119911)+119860

1198992(119909 119911)+119860

1198993(119909 119911)+119860

1198994(119909 119911)

11987811989911

(119909 119911)radic119881119899(119909 119911)rarr

119889

119873(0 1198682) by a triangular array version

of the Lindeberg-Levy central limit theorem The proof ofthe triangular-array version of the theorem is identical to theproof of the ordinary Lindeberg-Levy theorem The lemmafollows if we can prove that 119860

119899119895(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for

119895 = 2 3 4 and almost every (119909 119911) isin [0 1]2

Assumption 7 and arguments like those leading to (62)of HH [11] show that

∬

1

0

119881119899(119909 119911) 119889119909 119889119911 ≍ 119899

minus[2120573+120572minus120588(120572+1)](2120573+120572)

(A7)

It follows from the Cauchy-Schwartz inequality 119864(119891(minus119894)

XZW minus

119891XZW) = 119874(ℎ1015840

) and Var(119891(minus119894)

XZW) = 119874[1(119899ℎ2

)] that

11986410038171003817100381710038171198601198992

10038171003817100381710038172

= 119874(1

1198992ℎ21198862119899

+ℎ2119903

1198991198862119899

) (A8)

Therefore it follows from Assumptions 5 and 7 that119860

1198992(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909 119911) Now

consider 1198601198993(119909 119911) Define the operator Δ = minus 119879 Then

1198601198993

(119909 119911) = minus ( + 119886119899119868) Δ119860

1198991(119909 119911) (A9)

Therefore the Cauchy-Schwartz inequality gives

11986410038171003817100381710038171198601198992

10038171003817100381710038172

le 11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

119864100381710038171003817100381711987811989911

10038171003817100381710038172

= 11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

∬

1

0

119881119899(119909 119911) 119889119909 119889119911

(A10)

HH show that

11986410038171003817100381710038171003817( + 119886

119899119868) Δ

10038171003817100381710038171003817

2

= 119874(1

119899ℎ1198862119899

+ℎ2119903

1198862119899

) (A11)

Therefore it follows from Assumptions 5 and 7 that119860

1198993(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909 119911) Finally

some algebra shows that

1198601198994

(119909 119911) = minus( + 119886119899119868)

minus1

Δ1198601198992

(119909 119911) (A12)

Therefore 1198601198994(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1) for almost every (119909

119911) follows from (A11) and 1198601198992(119909 119911)radic119881

119899(119909 119911) = 119900

119901(1)

Lemma A2 (asymptotic negligibility of 1198781198992(119909 119911) minus 119892(119909 119911))

Let Assumptions 2ndash7 hold Then 1198781198992(119909 119911) minus 119892(119909 119911)

radic119881119899(119909 119911) = 119900

119901(1) for almost every (119909 119911)

Mathematical Problems in Engineering 9

Proof Define

119863119899(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times (119891XZW minus 119891XZW) (119909 119911 119908) 119889120579 119889120578 119889119908

1198601198991

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894 119885

119894) (119879

+

119891XZW) (119909 119911W119894)

(A13)

Redefine

1198601198992

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

minus 119863119899(119909 119911)

1198601198993

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) 119891XZW] (119909 119911W119894) + 119863

119899(119909 119911)

1198601198994

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) (119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894)

(A14)

Then 1198781198992(119909 119911) = sum

4

119895=1119860

119899119895(119909 119911) Arguments identical to

those used to derive (62) and (63) of HH [11] show that119864119860

1198991minus 119892

2

= 119874[119899minus120588(21205731)(2120573+120572)] and

∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874119899minus[2120573+120572minus120588(120572+1)](2120573+120572)

(A15)

Therefore it follows from Assumptions 5 and 7 that

[1198641198601198991

(119909 119911) minus 119892 (119909 119911)]

radic119881119899(119909 119911)

= 119900 (1) (A16)

119881minus1

119899(119909 119911)∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874 (1) (A17)

for almost every (119909 119911)Now consider 119860

1198992(119909 119911) Define

119863119899119894(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times(119891(minus119894)

XZWminus119891XZW)(119909 119911 119908) 119889120579 119889120578 119889119908

11986011989921

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894) minus 119863

119899119894(119909 119911)

(A18)

and 11986011989922

(119909 119911) = 119899minus1

sum119899

119894=1[119863

119899119894(119909 119911) minus 119863

119899(119909 119911)] HH show

that 11986411986011989921

2

= 119874((ℎ2119903

1198991198862

119899) + (1119899

2

ℎ2

1198862

119899)) and 119864119860

119899222

=

119874(11198992

1198862

119899) Therefore it follows from Assumptions 5 and 7

that

1198601198992

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A19)

for almost every (119909 119911) Now consider 1198601198993(119909 119911) Write

1198601198993

(119909 119911) = 11986011989931

(119909 119911) + 11986011989932

(119909 119911) (A20)

where 11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ119892(119909 119911) + 119863119899(119909 119911) and

11986011989932

(119909 119911) = minus(+

+ 119886119899119868)

minus1

Δ(1198601198991

minus 119892)(119909 119911) It follows from(A11)-(A16) and (A20) that

11986011989932

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A21)

for almost every (119909 119911)To analyze 119860

11989931(119909 119911) define

1198611198991

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

1198611198992

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119909 119908) 119892 (119909 119909) 119889119909 119889119911 119889119908

1198611198993

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)

119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989911

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989912

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989921

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989922

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

(A22)

10 Mathematical Problems in Engineering

Define 120575 = ℎ2119903

+ (119899ℎ)minus1 HH show that

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

(11986111989921

+ 11986111989922

) (119909 119911)

(A23)

Define

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

11986111989921

(A24)

Then

119864100381710038171003817100381711986011989931

10038171003817100381710038172

le const [1198641003817100381710038171003817100381711986011989931

10038171003817100381710038171003817

2

+ 11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

]

(A25)

11986410038171003817100381710038171003817119860

11989931

10038171003817100381710038171003817

2

le const (1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

11986111989912

1003817100381710038171003817

4

+1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

(A26)

HH show that

1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817 = 119874(ℎ119903

119886119899

) (A27)

(1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

)12

= 119874(120575ℎ

2119903

119886119899

) (A28)

(1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

= 119874(1205752

1198862119899

) (A29)

See (611) (613) (614) and (615) of HH [11] Moreover

11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

= 119874(ℎ2119903minus1

1198991198862+(120572+1)120572

119899

+1

1198993ℎ51198864119899

+ℎ4119903

119899ℎ1198862119899

)

(A30)

See the arguments leading to (624) in HH [11] and theanalogous result for their equation (624) in HH [11] andthe analogous result for their quantity 119864119867

11989922 Combining

(A25)ndash(A30) with Assumptions 5 and 7 yields the result that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

=minus(119868 + 119879

+

Δ)minus1

119879+

11986111989912

radic119881119899(119909 119911)

+ 119900119901(1) (A31)

Now consider minus(119868 +119879+

Δ)minus1

119879+

11986111989912

Standard calculations forkernel estimators show that

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 119899minus1

119899

sum

119894=1

119891XZW (119909 119911W119894) 119892 (X

119894Z

119894) + 119874 (ℎ

119903

)

(A32)

Therefore

119879+

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 1198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

)

(A33)

119879+

11986111989912

(119909 119911) = 1198601198991

(119909 119911) minus 1198641198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

) (A34)

But

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911)

= 119879+

11986111989912

+ [(119868 + 119879+

Δ)minus1

minus 119868]119879+

11986111989912

= 119879+

11986111989912

+ ( + 119886119899119868)

minus1

Δ119879+

11986111989912

(A35)

Therefore it follows by combining Assumption 7 and equa-tions (A11) (A17) and (A34) that

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911) = 1198601198991

(119911) minus 1198641198601198991

(119909 119911) + 119903119899

(A36)

where 1198641199031198992

radic119881119899(119909 119911) = 119900(1) for almost every (119909 119911)

Combining this result with (A21) and (A31) gives

1198601198993

(119909 119911)

radic119881119899(119909 119911)

=minus [119860

1198991(119909 119911) minus 119864119860

1198991(119909 119911)]

radic119881119899(119909 119911)

+ 119900119901(1) (A37)

for almost every (119909 119911)Now consider 119860

1198994(119909 119911) HH show that

1198601198994

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ (1198601198992

minus 119879+

1198611198992) (119909 119911)

(A38)

Therefore it follows from (A19) and (A30) that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A39)

for almost every (119909 119911)Now combine (A19) (A37) and (A39) to obtain

1198781198992

(119909 119911)

radic119881119899(119909 119911)

=sum4

119895=1119860

119899119895(119909 119911)

radic119881119899(119909 119911)

=119864119860

1198991(119909 119911)

radic119881119899(119909 119911)

+ 119900119901(1)

(A40)

for almost every (119909 119911)The lemma follows by combining thisresult with (A16)

This completes the proof

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 no 1 pp 141ndash1831973

[3] F Antonelli ldquoBackward-forward stochastic differential equa-tionsrdquo The Annals of Applied Probability vol 3 no 3 pp 777ndash793 1993

[4] HWangW Li and XWang ldquoAsymptotic stabilization by statefeedback for a class of stochastic nonlinear systems with time-varying coefficientsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 258093 6 pages 2014

[5] W Zhang and G Li ldquoDiscrete-time indefinite stochastic linearquadratic optimal control with second moment constraintsrdquoMathematical Problems in Engineering vol 2014 Article ID278142 9 pages 2014

[6] Y Su and L Lin ldquoSemi-parametric estimation for forward-backward stochastic differential equationsrdquo Communications inStatistics Theory and Methods vol 38 no 11 pp 1759ndash17752009

[7] X Chen and L Lin ldquoNonparametric estimation for FBS-DEs models with applications in financerdquo Communications inStatisticsmdashTheory and Methods vol 39 no 14 pp 2492ndash25142010

[8] L Lin F Li and L X Zhu ldquoOn regressionwith variance built-inmean regression function a new financial modelrdquo Manuscript2009

[9] Q Zhang and L Lin ldquoTerminal-dependent statistical inferencesfor FBSDErdquo Stochastic Analysis and Applications vol 32 pp128ndash151 2014

[10] C Hsiao Analysis of Panel Data vol 36 of Econometric SocietyMonographs Cambridge University Press Cambridge UK 2ndedition 2003

[11] P Hall and J L Horowitz ldquoNonparametric methods for infer-ence in the presence of instrumental variablesrdquo The Annals ofStatistics vol 33 no 6 pp 2904ndash2929 2005

[12] A N Kolmogorov and U A Rozanov ldquoOn the strong mixingconditions of a stationary Gaussian processrdquo Theory of Proba-bility and Its Applications vol 2 pp 222ndash227 1960

[13] R C Bradley and W Bryc ldquoMultilinear forms and measures ofdependence between random variablesrdquo Journal of MultivariateAnalysis vol 16 no 3 pp 335ndash367 1985

[14] C R Lu and Z Y Lin Limit Theories for Mixing DependentVariables Science Press Beijing China 1997

[15] D N Politis and J P Romano ldquoA general resampling scheme fortriangular arrays of120572-mixing randomvariableswith applicationto the problem of spectral density estimationrdquo The Annals ofStatistics vol 20 no 4 pp 1985ndash2007 1992

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Mathematical Problems in Engineering

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Proof Define

119863119899(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times (119891XZW minus 119891XZW) (119909 119911 119908) 119889120579 119889120578 119889119908

1198601198991

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894 119885

119894) (119879

+

119891XZW) (119909 119911W119894)

(A13)

Redefine

1198601198992

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)] (119909 119911W119894)

minus 119863119899(119909 119911)

1198601198993

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) 119891XZW] (119909 119911W119894) + 119863

119899(119909 119911)

1198601198994

(119909 119911)

= 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [(

+

minus 119879+

) (119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894)

(A14)

Then 1198781198992(119909 119911) = sum

4

119895=1119860

119899119895(119909 119911) Arguments identical to

those used to derive (62) and (63) of HH [11] show that119864119860

1198991minus 119892

2

= 119874[119899minus120588(21205731)(2120573+120572)] and

∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874119899minus[2120573+120572minus120588(120572+1)](2120573+120572)

(A15)

Therefore it follows from Assumptions 5 and 7 that

[1198641198601198991

(119909 119911) minus 119892 (119909 119911)]

radic119881119899(119909 119911)

= 119900 (1) (A16)

119881minus1

119899(119909 119911)∬

1

0

Var [1198601198991

(119909 119911)] 119889119909 119889119911 = 119874 (1) (A17)

for almost every (119909 119911)Now consider 119860

1198992(119909 119911) Define

119863119899119894(119909 119911) = ∭

1

0

119892 (120579 120578) 119891XZW (120579 120578 119908) 119879+

times(119891(minus119894)

XZWminus119891XZW)(119909 119911 119908) 119889120579 119889120578 119889119908

11986011989921

(119909 119911) = 119899minus1

119899

sum

119894=1

119892 (X119894Z

119894) [119879

+

(119891(minus119894)

XZW minus 119891XZW)]

times (119909 119911W119894) minus 119863

119899119894(119909 119911)

(A18)

and 11986011989922

(119909 119911) = 119899minus1

sum119899

119894=1[119863

119899119894(119909 119911) minus 119863

119899(119909 119911)] HH show

that 11986411986011989921

2

= 119874((ℎ2119903

1198991198862

119899) + (1119899

2

ℎ2

1198862

119899)) and 119864119860

119899222

=

119874(11198992

1198862

119899) Therefore it follows from Assumptions 5 and 7

that

1198601198992

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A19)

for almost every (119909 119911) Now consider 1198601198993(119909 119911) Write

1198601198993

(119909 119911) = 11986011989931

(119909 119911) + 11986011989932

(119909 119911) (A20)

where 11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ119892(119909 119911) + 119863119899(119909 119911) and

11986011989932

(119909 119911) = minus(+

+ 119886119899119868)

minus1

Δ(1198601198991

minus 119892)(119909 119911) It follows from(A11)-(A16) and (A20) that

11986011989932

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A21)

for almost every (119909 119911)To analyze 119860

11989931(119909 119911) define

1198611198991

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

1198611198992

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119909 119908) 119892 (119909 119909) 119889119909 119889119911 119889119908

1198611198993

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)

119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989911

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989912

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989921

(119909 119911) = ∭

1

0

[119864119891XZW (119909 119911 119908) minus 119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

11986111989922

(119909 119911) = ∭

1

0

[119891XZW (119909 119911 119908) minus 119864119891XZW (119909 119911 119908)]

times 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

(A22)

10 Mathematical Problems in Engineering

Define 120575 = ℎ2119903

+ (119899ℎ)minus1 HH show that

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

(11986111989921

+ 11986111989922

) (119909 119911)

(A23)

Define

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

11986111989921

(A24)

Then

119864100381710038171003817100381711986011989931

10038171003817100381710038172

le const [1198641003817100381710038171003817100381711986011989931

10038171003817100381710038171003817

2

+ 11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

]

(A25)

11986410038171003817100381710038171003817119860

11989931

10038171003817100381710038171003817

2

le const (1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

11986111989912

1003817100381710038171003817

4

+1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

(A26)

HH show that

1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817 = 119874(ℎ119903

119886119899

) (A27)

(1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

)12

= 119874(120575ℎ

2119903

119886119899

) (A28)

(1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

= 119874(1205752

1198862119899

) (A29)

See (611) (613) (614) and (615) of HH [11] Moreover

11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

= 119874(ℎ2119903minus1

1198991198862+(120572+1)120572

119899

+1

1198993ℎ51198864119899

+ℎ4119903

119899ℎ1198862119899

)

(A30)

See the arguments leading to (624) in HH [11] and theanalogous result for their equation (624) in HH [11] andthe analogous result for their quantity 119864119867

11989922 Combining

(A25)ndash(A30) with Assumptions 5 and 7 yields the result that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

=minus(119868 + 119879

+

Δ)minus1

119879+

11986111989912

radic119881119899(119909 119911)

+ 119900119901(1) (A31)

Now consider minus(119868 +119879+

Δ)minus1

119879+

11986111989912

Standard calculations forkernel estimators show that

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 119899minus1

119899

sum

119894=1

119891XZW (119909 119911W119894) 119892 (X

119894Z

119894) + 119874 (ℎ

119903

)

(A32)

Therefore

119879+

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 1198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

)

(A33)

119879+

11986111989912

(119909 119911) = 1198601198991

(119909 119911) minus 1198641198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

) (A34)

But

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911)

= 119879+

11986111989912

+ [(119868 + 119879+

Δ)minus1

minus 119868]119879+

11986111989912

= 119879+

11986111989912

+ ( + 119886119899119868)

minus1

Δ119879+

11986111989912

(A35)

Therefore it follows by combining Assumption 7 and equa-tions (A11) (A17) and (A34) that

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911) = 1198601198991

(119911) minus 1198641198601198991

(119909 119911) + 119903119899

(A36)

where 1198641199031198992

radic119881119899(119909 119911) = 119900(1) for almost every (119909 119911)

Combining this result with (A21) and (A31) gives

1198601198993

(119909 119911)

radic119881119899(119909 119911)

=minus [119860

1198991(119909 119911) minus 119864119860

1198991(119909 119911)]

radic119881119899(119909 119911)

+ 119900119901(1) (A37)

for almost every (119909 119911)Now consider 119860

1198994(119909 119911) HH show that

1198601198994

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ (1198601198992

minus 119879+

1198611198992) (119909 119911)

(A38)

Therefore it follows from (A19) and (A30) that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A39)

for almost every (119909 119911)Now combine (A19) (A37) and (A39) to obtain

1198781198992

(119909 119911)

radic119881119899(119909 119911)

=sum4

119895=1119860

119899119895(119909 119911)

radic119881119899(119909 119911)

=119864119860

1198991(119909 119911)

radic119881119899(119909 119911)

+ 119900119901(1)

(A40)

for almost every (119909 119911)The lemma follows by combining thisresult with (A16)

This completes the proof

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 no 1 pp 141ndash1831973

[3] F Antonelli ldquoBackward-forward stochastic differential equa-tionsrdquo The Annals of Applied Probability vol 3 no 3 pp 777ndash793 1993

[4] HWangW Li and XWang ldquoAsymptotic stabilization by statefeedback for a class of stochastic nonlinear systems with time-varying coefficientsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 258093 6 pages 2014

[5] W Zhang and G Li ldquoDiscrete-time indefinite stochastic linearquadratic optimal control with second moment constraintsrdquoMathematical Problems in Engineering vol 2014 Article ID278142 9 pages 2014

[6] Y Su and L Lin ldquoSemi-parametric estimation for forward-backward stochastic differential equationsrdquo Communications inStatistics Theory and Methods vol 38 no 11 pp 1759ndash17752009

[7] X Chen and L Lin ldquoNonparametric estimation for FBS-DEs models with applications in financerdquo Communications inStatisticsmdashTheory and Methods vol 39 no 14 pp 2492ndash25142010

[8] L Lin F Li and L X Zhu ldquoOn regressionwith variance built-inmean regression function a new financial modelrdquo Manuscript2009

[9] Q Zhang and L Lin ldquoTerminal-dependent statistical inferencesfor FBSDErdquo Stochastic Analysis and Applications vol 32 pp128ndash151 2014

[10] C Hsiao Analysis of Panel Data vol 36 of Econometric SocietyMonographs Cambridge University Press Cambridge UK 2ndedition 2003

[11] P Hall and J L Horowitz ldquoNonparametric methods for infer-ence in the presence of instrumental variablesrdquo The Annals ofStatistics vol 33 no 6 pp 2904ndash2929 2005

[12] A N Kolmogorov and U A Rozanov ldquoOn the strong mixingconditions of a stationary Gaussian processrdquo Theory of Proba-bility and Its Applications vol 2 pp 222ndash227 1960

[13] R C Bradley and W Bryc ldquoMultilinear forms and measures ofdependence between random variablesrdquo Journal of MultivariateAnalysis vol 16 no 3 pp 335ndash367 1985

[14] C R Lu and Z Y Lin Limit Theories for Mixing DependentVariables Science Press Beijing China 1997

[15] D N Politis and J P Romano ldquoA general resampling scheme fortriangular arrays of120572-mixing randomvariableswith applicationto the problem of spectral density estimationrdquo The Annals ofStatistics vol 20 no 4 pp 1985ndash2007 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Define 120575 = ℎ2119903

+ (119899ℎ)minus1 HH show that

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

(11986111989921

+ 11986111989922

) (119909 119911)

(A23)

Define

11986011989931

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

(11986111989911

+ 11986111989912

+ 1198611198993) (119909 119911)

+ (119868 + 119879+

Δ)minus1

119879+

Δ119879+

11986111989921

(A24)

Then

119864100381710038171003817100381711986011989931

10038171003817100381710038172

le const [1198641003817100381710038171003817100381711986011989931

10038171003817100381710038171003817

2

+ 11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

]

(A25)

11986410038171003817100381710038171003817119860

11989931

10038171003817100381710038171003817

2

le const (1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

11986111989912

1003817100381710038171003817

4

+1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

+ 1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

(A26)

HH show that

1003817100381710038171003817119879+

11986111989911

1003817100381710038171003817 = 119874(ℎ119903

119886119899

) (A27)

(1198641003817100381710038171003817119879

+

Δ119879+

11986111989921

1003817100381710038171003817

4

)12

= 119874(120575ℎ

2119903

119886119899

) (A28)

(1198641003817100381710038171003817119879

+

1198611198993

1003817100381710038171003817

4

)12

= 119874(1205752

1198862119899

) (A29)

See (611) (613) (614) and (615) of HH [11] Moreover

11986410038171003817100381710038171003817(119868 + 119879Δ)

minus1

119879+

Δ119879+

11986111989922

10038171003817100381710038171003817

2

= 119874(ℎ2119903minus1

1198991198862+(120572+1)120572

119899

+1

1198993ℎ51198864119899

+ℎ4119903

119899ℎ1198862119899

)

(A30)

See the arguments leading to (624) in HH [11] and theanalogous result for their equation (624) in HH [11] andthe analogous result for their quantity 119864119867

11989922 Combining

(A25)ndash(A30) with Assumptions 5 and 7 yields the result that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

=minus(119868 + 119879

+

Δ)minus1

119879+

11986111989912

radic119881119899(119909 119911)

+ 119900119901(1) (A31)

Now consider minus(119868 +119879+

Δ)minus1

119879+

11986111989912

Standard calculations forkernel estimators show that

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 119899minus1

119899

sum

119894=1

119891XZW (119909 119911W119894) 119892 (X

119894Z

119894) + 119874 (ℎ

119903

)

(A32)

Therefore

119879+

∭

1

0

119891XZW (119909 119911 119908) 119891XZW (119909 119911 119908) 119892 (119909 119911) 119889119909 119889119911 119889119908

= 1198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

)

(A33)

119879+

11986111989912

(119909 119911) = 1198601198991

(119909 119911) minus 1198641198601198991

(119909 119911) + 119900 (ℎ119903

119886119899

) (A34)

But

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911)

= 119879+

11986111989912

+ [(119868 + 119879+

Δ)minus1

minus 119868]119879+

11986111989912

= 119879+

11986111989912

+ ( + 119886119899119868)

minus1

Δ119879+

11986111989912

(A35)

Therefore it follows by combining Assumption 7 and equa-tions (A11) (A17) and (A34) that

(119868 + 119879+

Δ)minus1

119879+

11986111989912

(119909 119911) = 1198601198991

(119911) minus 1198641198601198991

(119909 119911) + 119903119899

(A36)

where 1198641199031198992

radic119881119899(119909 119911) = 119900(1) for almost every (119909 119911)

Combining this result with (A21) and (A31) gives

1198601198993

(119909 119911)

radic119881119899(119909 119911)

=minus [119860

1198991(119909 119911) minus 119864119860

1198991(119909 119911)]

radic119881119899(119909 119911)

+ 119900119901(1) (A37)

for almost every (119909 119911)Now consider 119860

1198994(119909 119911) HH show that

1198601198994

(119909 119911) = minus(119868 + 119879+

Δ)minus1

119879+

Δ (1198601198992

minus 119879+

1198611198992) (119909 119911)

(A38)

Therefore it follows from (A19) and (A30) that

1198601198994

(119909 119911)

radic119881119899(119909 119911)

= 119900119901(1) (A39)

for almost every (119909 119911)Now combine (A19) (A37) and (A39) to obtain

1198781198992

(119909 119911)

radic119881119899(119909 119911)

=sum4

119895=1119860

119899119895(119909 119911)

radic119881119899(119909 119911)

=119864119860

1198991(119909 119911)

radic119881119899(119909 119911)

+ 119900119901(1)

(A40)

for almost every (119909 119911)The lemma follows by combining thisresult with (A16)

This completes the proof

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 no 1 pp 141ndash1831973

[3] F Antonelli ldquoBackward-forward stochastic differential equa-tionsrdquo The Annals of Applied Probability vol 3 no 3 pp 777ndash793 1993

[4] HWangW Li and XWang ldquoAsymptotic stabilization by statefeedback for a class of stochastic nonlinear systems with time-varying coefficientsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 258093 6 pages 2014

[5] W Zhang and G Li ldquoDiscrete-time indefinite stochastic linearquadratic optimal control with second moment constraintsrdquoMathematical Problems in Engineering vol 2014 Article ID278142 9 pages 2014

[6] Y Su and L Lin ldquoSemi-parametric estimation for forward-backward stochastic differential equationsrdquo Communications inStatistics Theory and Methods vol 38 no 11 pp 1759ndash17752009

[7] X Chen and L Lin ldquoNonparametric estimation for FBS-DEs models with applications in financerdquo Communications inStatisticsmdashTheory and Methods vol 39 no 14 pp 2492ndash25142010

[8] L Lin F Li and L X Zhu ldquoOn regressionwith variance built-inmean regression function a new financial modelrdquo Manuscript2009

[9] Q Zhang and L Lin ldquoTerminal-dependent statistical inferencesfor FBSDErdquo Stochastic Analysis and Applications vol 32 pp128ndash151 2014

[10] C Hsiao Analysis of Panel Data vol 36 of Econometric SocietyMonographs Cambridge University Press Cambridge UK 2ndedition 2003

[11] P Hall and J L Horowitz ldquoNonparametric methods for infer-ence in the presence of instrumental variablesrdquo The Annals ofStatistics vol 33 no 6 pp 2904ndash2929 2005

[12] A N Kolmogorov and U A Rozanov ldquoOn the strong mixingconditions of a stationary Gaussian processrdquo Theory of Proba-bility and Its Applications vol 2 pp 222ndash227 1960

[13] R C Bradley and W Bryc ldquoMultilinear forms and measures ofdependence between random variablesrdquo Journal of MultivariateAnalysis vol 16 no 3 pp 335ndash367 1985

[14] C R Lu and Z Y Lin Limit Theories for Mixing DependentVariables Science Press Beijing China 1997

[15] D N Politis and J P Romano ldquoA general resampling scheme fortriangular arrays of120572-mixing randomvariableswith applicationto the problem of spectral density estimationrdquo The Annals ofStatistics vol 20 no 4 pp 1985ndash2007 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 no 1 pp 141ndash1831973

[3] F Antonelli ldquoBackward-forward stochastic differential equa-tionsrdquo The Annals of Applied Probability vol 3 no 3 pp 777ndash793 1993

[4] HWangW Li and XWang ldquoAsymptotic stabilization by statefeedback for a class of stochastic nonlinear systems with time-varying coefficientsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 258093 6 pages 2014

[5] W Zhang and G Li ldquoDiscrete-time indefinite stochastic linearquadratic optimal control with second moment constraintsrdquoMathematical Problems in Engineering vol 2014 Article ID278142 9 pages 2014

[6] Y Su and L Lin ldquoSemi-parametric estimation for forward-backward stochastic differential equationsrdquo Communications inStatistics Theory and Methods vol 38 no 11 pp 1759ndash17752009

[7] X Chen and L Lin ldquoNonparametric estimation for FBS-DEs models with applications in financerdquo Communications inStatisticsmdashTheory and Methods vol 39 no 14 pp 2492ndash25142010

[8] L Lin F Li and L X Zhu ldquoOn regressionwith variance built-inmean regression function a new financial modelrdquo Manuscript2009

[9] Q Zhang and L Lin ldquoTerminal-dependent statistical inferencesfor FBSDErdquo Stochastic Analysis and Applications vol 32 pp128ndash151 2014

[10] C Hsiao Analysis of Panel Data vol 36 of Econometric SocietyMonographs Cambridge University Press Cambridge UK 2ndedition 2003

[11] P Hall and J L Horowitz ldquoNonparametric methods for infer-ence in the presence of instrumental variablesrdquo The Annals ofStatistics vol 33 no 6 pp 2904ndash2929 2005

[12] A N Kolmogorov and U A Rozanov ldquoOn the strong mixingconditions of a stationary Gaussian processrdquo Theory of Proba-bility and Its Applications vol 2 pp 222ndash227 1960

[13] R C Bradley and W Bryc ldquoMultilinear forms and measures ofdependence between random variablesrdquo Journal of MultivariateAnalysis vol 16 no 3 pp 335ndash367 1985

[14] C R Lu and Z Y Lin Limit Theories for Mixing DependentVariables Science Press Beijing China 1997

[15] D N Politis and J P Romano ldquoA general resampling scheme fortriangular arrays of120572-mixing randomvariableswith applicationto the problem of spectral density estimationrdquo The Annals ofStatistics vol 20 no 4 pp 1985ndash2007 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of