Stochastic Calc; Finance

8/12/2019 Stochastic Calc; Finance

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MIDDLE EAST TECHNICAL UNIVERSITY Ankara, Turkey

INSTITUTE OF APPLIED MATHEMATICS

http://iam.metu.edu.tr

Ito-Taylor Expansions for Systems ofStochastic Differential Equations with

Numerical ApplicationsFikriye Ylmaz 1 , Hacer Oz 2 and Gerhard Wilhelm Weber 3

Abstract. Stochastic differential equations (SDEs) are playing a growing role in

financial mathematics, actuarial sciences, physics, biology and engineering. In this

paper, we focus on a numerical simulation of systems of SDEs based on the stochas-

tic Taylor series expansions. At first, we apply the vector-valued It o formula to the

systems of SDEs, then, the stochastic Taylor formula is used to get the numerical

schemes. In the case of higher dimensional stochastic processes and equations, the

numerical schemes may be expensive and take more time to compute. Iterated

integrals with several n-dimensional Brownian motions can be quite complicated.

Sometimes, such integrals may be approximated. In this paper, we deal with both

the systems with standard n-dimensional Brownian motions and the systems of

SDEs having correlated Brownian motions. The main issue is to transform the

systems of SDEs with correlated Brownian motions to the ones having independent

Wiener processes, and then, to apply the Ito formula to the transformed systems.

Moreover, explicit computations of multiple Ito integrals with several Brownian

motions are given, supported by MATLAB. The paper ends with conclusion and

outlook to future studies.

Keywords. Systems of SDEs, Ito-Taylor expansions, Correlated Brownianmotions, Vector-valued Ito formula, Numerical schemes for SDEs.

Preprint No. 2013-14

March 2013

1Gazi University, 06500 Ankara, Turkey. [email protected] of Applied Mathematics, Middle East Technical University, Ankara, Turkey.

[email protected] of Applied Mathematics, Middle East Technical University, Ankara, Turkey.

[email protected]


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1 Introduction

There has been a great interest in the simulation methods of SDEs in thefields of, e.g., finance, insurance, medicine and the modern technologies[10, 11, 16, 20, 21]. As the need to take into account of uncertainty is moreand more accepted in science and the applications, SDEs are an emerg-ing subject of interest. Stochastic Taylor expansion provides a source forthe discrete-time approximation methods. One of the simplest ways to dis-cretize the process is Euler method, which approximates the integrals byusing the left-point rule. TheMilsteinscheme (1974), which has the order1.0 of strong convergence, is stronger than Euler method. By adding fur-ther stochastic integrals, with the equations and using the stochastic Taylor

expansion, more accurate schemes can be obtained.

P.E. Kloeden and E. Platen [10] have given a methodical means of derivingthe Taylor series for both Stratonovich and Ito form of a SDE. The applica-tion of Ito Taylor formula to the 1-dimensional SDEs is given explicitly in[10]. In this case, Ii1,i2,...,ik represent the Ito integral, where integration iswith respect to ds ifik = 0, or dWs ifik = 1 [22]. For example,

I010=

t0

s10

s20

ds3dWs2ds1.

Such a display of integrals can be considered as a part of the inner beauty of

stochastic dynamics. This beauty expresses itself in terms of digitalization,algebraization and automization which are not only very aesthetic indeed,but also very practical.

By recursively using the Ito formula, the obtained Taylor series can be re-lated to a tree theory. The tree expansion is given for the true solution in[3]. Runge-Kutta method has been constructed, having order 1.5, in [3, 4, 5].

In the deterministic case, the accuracy of the numerical scheme can be ob-tained by comparing the obtained result with the exact solution. In thestochastic differential equations, there are two ways to measure the accuracy

of the solution: strong and weak convergence. A time-discrete approxima-tion Xh is said to converge stronglywith order p >0 at time Tif there is apositive constantc, independent ofh, and h0> 0 such that

E(|XT Xh(T)|) chp, h (0, h0).

On the other hand, Xh is said to converge weaklywith order p >0 at timeTif there are constants c >0, independent ofh, and h0 > 0 such that

|E(XT) E(Xh(T))| chp, h (0, h0).

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In this paper, in order to get numerical solutions, we shall consider Taylor

schemes that converge strongly.

Vector-valued Ito calculus may involve more than one Wiener process. Al-though it seems that the extension of one-variable SDEs to the multi-valuedSDEs is easy, the computations of iterated Ito integrals are very expensivein the numerical approximations. But, the systems of SDEs model the im-portant cases having several source of randomness and correlation; these aremain reasons of financial risk and instability.

There are not many packages with regards to the simulations of SDEs [7].The Maple package stochasticprovides a symbolic manipulation for SDEs.

The package may be downloaded fromwww.math.uni-frankfurt.de/numerik/kloeden. Many numerical schemes may be generated by using this package.However, the computations of the iterated multi-dimensional Ito integralsare not supported. The software package SDE lab generated by H. Gilsingand T. Shardlow [6] is a good source especially for 1-dimensional SDEs.

In this work, we provide a routine, supported by MATLAB, to compute theiterated multi-dimensional Ito integrals with presence of the formulationsgiven in [10]. We use the Polar Marsaglia methodto generate random vari-ables. This method, that is attributed to G. Marsaglia, is a variation ofthe Box-Muller method. It is based on choosing random points (x, y) in the

square given by1< x


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ij =

1 ifi = j,0 ifi =j.

We shall call the process Wt:= (W1t, W2t, . . . , W

nt )

T as a correlated Brown-ian motion if

dWit dWjt =ijdt, for i, j= 1, 2, . . . , n ,

for a positive symmetric matrix = (ij)1i,jn satisfying the follows:

ii= 1, and ij =ji [1, 1] for i =j.The paper is organized as follows. Multivariable Ito calculus is reviewed

in Section 2. In Section 3, the systems of SDEs with independent Wienerprocesses and their Taylor expansions are analyzed. Then, the correlatedsystems are covered in Section 4. After obtaining discretization schemesin Section 5, the numerical results and implementation issues are given inSection 6. We conclude and give an outlook to the future studies in Section7.

2 Multi-dimensional Ito Calculus

We consider the process Xt in Rd. Let Zt be a multi-dimensional Brownian

motion, defined asZ

t= (Z

1

t, Z

2

t, . . . , Z

n

t)

T

. Then, the k

th

component of thevector-valued SDE is given by

dXkt =akdt+n

j=1

hkjdZjt , k= 1, 2, . . . , d ,

where ak(t,Xt) and hkj(t,Xt) are the drift and the diffusion coefficients, re-spectively.

We define A(t,Xt) = (a1(t,Xt), . . . , ad(t,Xt))T, Xt= (X1t, . . . , X

dt)

T, and

H(t,Xt) =

h11(t,Xt) . . . h1n(t,Xt)...

. . . ...

hd1(t,Xt) . . . hdn(t,Xt)

,

to get the following matrix formulation:

dXt= Adt+ HdZt. (1)

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3 Ito-Taylor approximation for standard Brown-

ian Motions

In this section, we will assume that the Brownian motions are independent.Eqn. (1) can be written in integral form as

Xt= Xt0+

tt0

A(s,Xs)ds+

tt0

H(s,Xs)dZs. (2)

The second integral is called Ito stochastic integral, which is defined by K.Ito in 1940 [9]. This integral can be approximated by stochastic Taylormethod. Before going into the numerical schemes, we recall Ito Lemma in

several dimensions.

Theorem 3.1 (Ito Lemma in several dimensions, [18])LetXt = (X1t, . . . , X

dt)

T be a vector-valued Ito process satisfying Eqn. (1).Letg : [0,) Rd Rp be a given bounded function in C2([0,) Rd).Then,

dg(t,Xt) =g

tdt+

di=1

g

xidXit +

1

2

di,j=1

2g

xi, xjdXitdX

jt . (3)

We consider now the k th component of the system of SDEs (1):

dXkt =ak(t,Xt)dt+n

j=1

hkj(t,Xt)dZjt ,

where ak(t,Xt) =ak(t, X1t, . . . , X

dt) and hkj(t,Xt) =hkj(t, X

1t, . . . , X

dt).

For applying the Ito formula, we let g(t,Xt) = (g1(t,Xt)), . . . , gp(t,Xt))T

and Yt= g(t,Xt).

Then, the component Ykt is given by

dYkt =gk

t dt+

di=1

gk

xidXit+

1

2

di,j=1

2gk

xixjdXitdX

jt .

Equivalently,

dYkt =

gk

t +

di=1

aigk

xi+

1

2

di,j=1

np=1

hjphip2gk

xixj

dt +

dj=1

np=1

hjpgk

xjdZpt,

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where all derivatives ofgk are to be evaluated in (t,Xt) and Brownian mo-

tions are independent.

The system of SDEs can be classified with respect to the shared states andBrownian motions. We give explicit formulations of each classification inthe following subsections.

3.1 Completely decoupled systems

Now, we apply the theorem to

dXkt =ak(t, Xkt)dt+hkk(t, X

kt)dZ

kt, k= 1, 2, . . . , d ,

where all equations have their own independent Brownian motions and statesthat implies d = n. Then,

dYkt = (gk

t +

di=1

aigk

xi +

1

2

di=1

hiihii2gk

xixi)dt+

di=1

hiigk

xidZit ,

where gk(t,Xt) = (X1t, . . . , X dt)

T. Herewith,

dXkt =ak(t, Xkt)dt+hkk(t, X

kt)dZ

kt. (4)

The integral form of Eqn. (4) is

Xkt =Xkt0

+

tt0

ak(s, Xks )ds+

tt0

hkk(s, Xks )dZ

ks . (5)

Before obtaining the Taylor series expansions, we define the following oper-ators:

L0 :=

t+

di=1

ai

xi+

1

2

di,j=1

np=1

hjphip2

xixj,

and

Lj :=d

p=1

hpj

xp

, for j = 1, 2, . . . , n .

Application of Theorem 3.1 to Eqn. (5) gives

Ykt =Ykt0

+

tt0

L0gkds+n

j=1

tt0

LjgkdZjs . (6)

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In Eqn. (5), firstly, gk is chosen as ak(t, Xkt) and then, the Ito formula (6)

is applied. Secondly, we let gk =hkk(t, Xkt) and use the Ito formula to get

Xkt =Xkt0

+

tt0

ak(t0, X

kt0

) +

st0

L0ak(, Xk)d

+n

j=1

st0

Ljak(, Xk)dZj

ds+

tt0

hkk(t0, X

kt0

)

+

st0

L0hkk(, Xk)d+n

j=1

st0

Ljhkk(, Xk)dZj

dZks ,

=Xkt0 + ak(t0, Xkt0

)I0+hkk(t0, Xkt0

)Ik

+

t

t0

st0

L0ak(, Xk)d+ nj=1

s

t0

Ljak(, Xk)dZj

ds (7)

+

tt0

st0

L0hkk(, Xk)d+n

j=1

st0

Ljhkk(, Xk)dZj

dZks .

We can continue with an application of Ito formula (6) to the functions L0ak,nj=1Ljak,L0hkkand

nj=1Ljhkkin (7), and then, to the functions L0L0ak,n

j=1LjL0ak,n

j=1L0Ljak,n

j,p=1LjLpak,L0L0hkk,n

j,p=1LjL0hkk,nj,p=1L0Ljhkk,

nj,p=1LjLphkk, to obtain Ito-Taylor expansion:

Xkt =Xkt0

+ ak(t0, Xkt0)I0+hkk(t0, Xkt0

)Ik

+ L0ak(t0, Xkt0)I00+n

j=1

Ljak(t0, Xkt0)Ij0

+ L0hkk(t0, Xkt0)I0k+n

j=1

Ljhkk(t0, Xkt0)Ijk

+ L0L0ak(t0, Xkt0)I000+n

j=1

LjL0ak(t0, Xkt0)Ij00

+

n

j=1

L0

Lj

ak(t0, Xk

t0)I0j0+

n

j,p=1

Lj

Lp

ak(t0, Xk

t0)Ijp0

+ L0L0hkk(t0, Xkt0)I00k+n

j,p=1

LjL0hkk(t0, Xkt0)Ij0k

+n

j,p=1

L0Ljhkk(t0, Xkt0)I0jk +n

j,p=1

LjLphkk(t0, Xkt0)Ijpk +R,

where R denotes the remainder term and Ii1,i2,...,ik represent the multiple

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Ito integral, which is defined as [10]:

I =

1, ifk = 0,tt0

Ids, ifk 1 and k = 0,tt0

IdZks , ifk 1 and k 1,

for = (1, 2, . . . , k)T (N0)k with k 2, where , the multi index,

can be obtained by deleting the last component of, i.e., integration is withrespect to ds ifik = 0, or dZ

js ifik = j= 0, j = 1, . . . , n.

3.2 Systems with common states

Now, we consider the following system

dXkt =ak(t,Xt) +hkk(t,Xt)dZkt, k= 1, 2, . . . , d , (8)

where all equations may have all states in common but they are not sharingBrownian motions.It has a similar formulation with completely decoupled systems

Xkt =Xkt0

+ ak(t0,Xt0)I0+hkk(t0,Xt0)Ik

+ L0ak(t0,Xt0)I00+n

j=1

Ljak(t0,Xt0)Ij0

+ L0hkk(t0,Xt0)I0k+n

j=1

Ljhkk(t0,Xt0)Ijk

+ L0L0ak(t0,Xt0)I000+n

j=1

LjL0ak(t0,Xt0)Ij00

+

nj=1

L0Ljak(t0,Xt0)I0j0+n

j,p=1

LjLpak(t0,Xt0)Ijp0

+

L0

L0hkk(t0,Xt0)I00k+

n

j,p=1L

j

L0hkk(t0,Xt0)Ij0k

+

nj,p=1

L0Ljhkk(t0,Xt0)I0jk +n

j,p=1

LjLphkk(t0,Xt0)Ijpk +R,

where R denotes the remainder term.

3.3 General case

We apply the Theorem 3.1 to the following system:

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dXkt =ak(t,Xt) +n

j=1

hkj(t,Xt)dZjt , k= 1, 2, . . . , d , (9)

where all equations may share both Brownian motions and states.

Then, similarly,

Xkt =Xkt0

+

tt0

ak(s,Xs)ds+n

j=1

tt0

hkj(s,Xs)dZjs . (10)

Again, we first choosegk :=ak(t,Xt) to apply the Ito formula, and then, welet gk :=hkj(t,Xt) to get

Xkt =Xkt0

+

tt0

ak(t0,Xt0) +

st0

L0ak(,X)d

+n

l,j=1

st0

Ljak(, X)dZj

ds+n

j=1

tt0

hkj(t0,Xt0)

+

st0

L0hkj(,X)d+n

l=1

st0

Llhkj(, X)dZl

dZjs ,

We do the same things as in the completely decoupled case, but only differ-ence is that ak and hkj depend on (t,Xt). Then,

Xkt =Xkt0

+ ak(t0,Xt0)I0+n

j=1

hkj(t0,Xt0)Ij

+ L0ak(t0,Xt0)I00+n

j=1

Ljak(t0,Xt0)Ij0

+n

j=1

L0hkj(t0,Xt0)I0j+n

l,j=1

Llhkj(t0,Xt0)Ilj

+ L

0

L0ak(t0,Xt0)I000+

n

j=1L

j

L0ak(t0,Xt0)Ij00 (11)

+n

j=1

L0Ljak(t0,Xt0)I0j0+n

j,p=1

LjLpak(t0,Xt0)Ijp0

+n

j=1

L0L0hkj(t0,Xt0)I00j+n

j,l=1

LlL0hkj(t0,Xt0)Il0j

+n

j,p,l=1

L0Llhkj(t0,Xt0)I0lj +n

j,p,l=1

LlLphkj(t0,Xt0)Ilpj+R.

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4 Ito-Taylor approximation for Correlated Brow-

nian Motions

In this section, we assume that Brownian motions are correlated, so weuse W instead ofZ in order to point out the difference from the standardBrownian motions. Now, we consider the system

dXkt =ak(t,Xt) +

n1

hkj(t,Xt)dWjt, k= 1, 2, . . . , d= n,

where dWit dWjt =ijdt.

Then, we write the correlation matrix as:

=

1 12 . . . 1n21 1 . . . 2n

... ...

. . . ...

n1 n2 . . . 1

, ij =ji [1, 1].

Here, is positive symmetric matrix which means that = T and

ni,j=1

ijxixj 0,

for all x= (x1, . . . , xn)T Rn.

By using standard Linear Algebra, one can find an n n matrix B =(bij)1i,jn such that

= BBT.

Moreover, using Cholesky Decomposion, we can take B as an upper (orlower) triangular matrix.

Correlated Brownian motions can be interpreted as

Wt= BZt,

where Wt = (W1t, . . . , W

nt )

T is a standard n-dimensional Brownian motionand the Brownian motions, Wit for i = 1, 2, . . . , n, are correlated.

In componentwise, notation,

Wit =n

j=1

bijZjt , i= 1, 2, . . . , n .

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Example 1. We consider 2-dimensional version ofweakly-coupled Ornstein-

Uhlenbeck model:

dX1t =1(1 X1t)dt+1dW1t,dX2t =2(2 X2t)dt+2dW2t, (12)

wheredW1tdW2t =dt,all coefficients are real and positive, and (1, 1).

Then, the correlation matrix can be written as:

=

1 1

=

1 0

1 2

1

0

1 2

,

by Cholesky Decomposition [2].

Thus,

Wt = BZt=

1 0

1 2

Z1tZ2t

.

So, Eqn. (12) becomes

dX1t =1(1 X1t)dt+1dZ1t,dX2t =2(2 X2t)dt+2dZ1t +2

1 2dZ2t. (13)

In integral form,

X1t =X1t0

+ 11

tt0

ds 1 tt0

X1s ds+1

tt0

dZ1s ,

X2t =X2t0

+ 22

tt0

ds 2 tt0

X2s ds+2

tt0

dZ1s +2

1 2 tt0

dZ2s .

Applying the procedure in the previous sections, we get

X1t =X1t0

+ 1(1 X1t0)I0+1I1 21(1 X1t0)I00 11I10+31(1 X1t0)I000+211I100+R,

X2t =X2t0 + 2(2 X2t0)I0+2I1+21 2I2 22(2 X2t0)I00 22I10 22

1 2I20

+ 32(2 X2t0)I000+222I100+

1 2222I200+R.

Example 2. We state the 2-dimensional version ofstrongly-coupled Ornstein-Uhlenbeck model

dX1t = (11X1t 12X2t)dt+1dW1t,dX2t = (21X1t 22X2t)dt+2dW2t, (14)

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where dW1tdW2t =dt and 1, 2= 0.

We obtain the following system in terms of standard Brownian motions:

dX1t = (11X1t 12X2t)dt+1dZ1t,dX2t = (21X1t 22X2t)dt+2dZ1t +2

1 2dZ2t. (15)

In a similar way, we obtain:

X1t =X1t0 (11X1t0+12X2t0)I0+1I1 (111+212)I10

+ 11(11X1t0

+ 12X2

t0) +12(21X

1

t0+22X

2

t0)I00

212

1 2I20

(211+1221)(11X1t0

+12X2t0

)

+ (1112+1222)(21X1t0

+ 22X2t0

)

I000

+

1(

211+1221) +2(1112+1222)

I100

+ 2(1112+1222)

1 2I200+R,

and

X2t =X2t0 (21X1t0+ 22X2t0)I0+2I1+2

1 2I2

(121+222)I10+

21(11X1t0

+12X2t0

)

+ 22(21X1t0

+ 22X2t0

)

I00 222

1 2I20 [(2111+2221)(11X1t0+12X2t0) + (2112+ 222)(21X

1t0

+ 22X2t0

)]I000+ [1(1121+2221)

+ 2(2112+222)]I100+2(1221+

222)

1 2I200+R.

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5 Discretization schemes with Strong Taylor ap-

proximations

We represent some numerical approximations of Ito integrals. Let denotethe increments, then:

Ij =

n+1n

dZjs = Zj =Zjn+1Zjn ,

Ij0=

n+1n

sn

dZjuds= W ,

I0j =

n+1n

sn

dudZjs = (Zj) W ,

Ijj = 12

(Zj)2 ,

Ijj0= I0jj =Ij0j =1

6

(Zj)2 ,Ijjj =

1

6

(Zj)3 3(Zj),Ij00= I0j0= I00j =

1

62Zjs ,

where the Wand Zj are Gaussian random variables with Zj N(0, ),W N(0, 1

33) and E

ZjW) = 1

22.

Now, we shall use the above relations to propose some strong approxima-tions.

5.1 The Euler-Maruyama Scheme

The simplest example of a strong Taylor approximation Y of the solutionof Eqn. (9) is the Euler-Maruyama or Euler method attaining the order ofstrong convergence 0.5. The kth component of the Euler scheme is of theform

Ykn+1= Ykn +ak +

nj=1

hkjZj, for k = 1, 2, . . . , d . (16)

In the cases where drift and difussion coefficients are nearly constant, thismethod generally gives us good numerical results. However, when the coeffi-cients are nonlinear the method can provide a poor estimate of the solution.So, higher-order schemes should be used to obtain more satisfactory schemes.

Let us consider the system (15) by taking 11 = 21 = 22 = 1, 12 =2, 1 = 2= 1 and = 0.6 to get

dX1t = (X1t 2X2t)dt+dZ1t,dX2t = (X1t X2t)dt+ 0.6dZ1t + 0.8dZ2t. (17)

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Euler scheme reads the system (17) as:

Y1n+1= Y1n + (Y2n 2Y1n ) + Z1,

Y2n+1= Y2n + (Y2n Y1n ) + 0.6Z1 + 0.8Z2.

5.2 The Milstein Scheme

Milstein scheme has the order of strong convergence 1.0. By including an ad-ditional term from Ito-Taylor expansion (11), we obtain theMilstein schemewhose k th component is of the form

Ykn+1= Ykn +ak +

n

j=1

hkjZj +

n

j1,j2=1

Lj1hkj2Ij1j2, for k = 1, 2, . . . , d .

(18)We note that Milstein scheme is identical to the Euler scheme when thediffusion term does not contain an Xt term.

Applying the scheme to the system (17), we obtain the Milstein scheme:

Y1n+1= Y1n + (Y2n 2Y1n ) + Z1,

Y2n+1= Y2n + (Y2n Y1n ) + 0.6Z1 + 0.8Z2.

5.3 The Order 1.5 Strong Taylor Scheme

We can get more accurate strong Taylor schemes by including further multi-ple stochastic integrals from the stochastic Taylor approximation (11). Thekth component ofthe order 1.5 strong Taylor schemeis given by

Ykn+1= Ykn + ak +

1

2L0ak

2 +n

j=1

(hkjZj +L0hkjI0j+L

jakIj0

+n

j1,j2=1

Lj1hkj2Ij1j2+n

j1,j2,j3=1

Lj1Lj2hkj3Ij1j2j3 (19)

for k = 1, 2, . . . , d .

Applying this method to the system (17), we get:

Y1n+1= Y1n + (Y2n 2Y1n ) +

1

2(3Y2n + 5Y

1n )

2 + Z1 2.6I10 0.8I20,

Y2n+1= Y2n + (Y2n Y1n ) +

1

2(2Y2n + 3Y

1n )

2 + 0.6Z1

+0.8Z2 1.6I10 0.8I20.

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5.4 The Order 2.0 Strong Taylor Scheme

The numerical results which was given by the order 2.0 strong Taylor schemeis better than other three method. Thekth component ofthe order 2.0 strongTaylor schemetakes the form

Ykn+1= Ykn + ak +

1

2L0ak

2 +n

j=1

(hkjZj +L0hkjI0j+L

jakIj0)

+

nj1,j2=1

Lj1hkj2Ij1j2+ L

0Lj1hkj2I0j1j2+Lj1L0hkj2Ij10j2

+ Lj1Lj2akIj1j20+n

j1,j2,j3=1

Lj1Lj2hkj3Ij1j2j3 (20)

+n

j1,j2,j3,j4=1

Lj1Lj2Lj2hkj4Ij1j2j3j4 ,

for k = 1, 2, . . . , d.

Finally, for the Eqn. (17) this formulation can be reduced to:

Y1n+1= Y1n + (Y2n 2Y1n ) +

1

2(3Y2n + 5Y

1n )

2 + Z1 2.6I10 0.8I20,

Y2n+1= Y

2n + (Y

2n Y

1n ) +

1

2 (2Y2n + 3Y

1n )

2

+ 0.6Z1

+0.8Z2 1.6I10 0.8I20.

6 Numerical Results and Implementation Details

In this section, we consider numerical examples for both the systems withindependent and correlated Brownian motions. In order to implement thediscrete scheme, we use MATLAB. There are many documentations thatdescribes the main features of MATLAB commands related to SDE. Somenumerical interpretations can be found in the SDEs MATLAB packages [7].

However, numerical examples implemented in MATLAB are mostly in the1-dimensional case. Some coupled SDEs are considered, but having sym-metric coefficients allowing easy computations that arise from multiple It ointegrals. Our first example demonstrates the triple SDEs having indepen-dent Brownian motions.

Example Run 1. The system of SDE consisting of three equationsproposed in Hofmann, Platen and Schweizer [8] is considered as:

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dX

1

t =X1

tX2

tdZ1

t,dX2t = (X2t X3t)dt+ 0.3X2tdZ2t,dX3t =

1

(X2t X3t)dt,

where X1t, X2t and X

3t represent the asset price, the instantenous volatil-

ity, and the averaged volatility, respectively. As in [7], Milstein scheme isobtained as:

dX1n+1= X1n + X

1nX

2nZ

1n+

1

2X1n(X

2n)

2{(Z1n)2 }

+ 0.3X1nX2n

tn+1

tn

t

tn

dZ2sdZ1s ,

dX2n+1= X2n (X2n X3n) + 0.3X2nZ2n+ 0.045X2n{(Z1n)2 },

dX3n+1= dX3n +

1

(X2n X3n).

We take = 1, X10 = 1, X20

= 0.1, X20 = 0.1 and T= 1; is considered as29.

The scheme has the double integraltn+1tn

ttn

dZ2sdZ1s . In [7], this integral is

approximated by Euler method. Although it is a bit challenging, we computesuch integrals by using the following formulations from [10] as follows:

Ip0

= , Ipj =

j, Ip00

= 1

22,

Ipj0=1

2

j+aj0

, Ip0j =

1

2

j aj0

.

Here,

aj0=

1

2

p

r=1

1

r

jr

2pjp ,

Ipj1j2 =1

2j1j2

1

2

(aj20j1 aj10j2) + Apj1j2,

Apj1j2 = 1

2

pr=1

1

r(j1rj2r j2rj1r),

with

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j = 1

Wj, jr =

2

rajr , jr =

2

rbjr ,

jp = 1

p

r=p+1

ajr , p = 1

12 1

22

pr=1

1

r2,

where j = 1, 2, . . . , mand r = 1, 2, . . . , p, for a positive number p chosen asfollows:

p= p() K2

,

for an appropriate positive constant K to ensure the convergence order ofthe numerical scheme.

Here, we note thatjr , jr and jp are independent Gaussian random vari-ables. We use the Polar Marsaglia Methodto generate the pairs of randomvariables.The following lines show the implementation of this method in MATLAB:

%Polar Marsaglia Method

function [z1,z2]= Polar

l=0.5;while l>0

u1 = rand;u2 = rand;

v1 = 2*u1 - 1;v2 = 2*u2 - 1;

V = (v1.*v1)+(v2.*v2);

if (V0)

break;

end

end

z1 = v1.*sqrt(-2*log(V)./V);

z2 = v2.*sqrt(-2*log(V)./V);

We can compute the iterated integraltn+1tn

ttn

dZ2s dZ1s as:

%Approximation of I_ij

function I_ij=ito_ij(p,Delta,G1,G2,mu1_j,mu2_j,ro)

a_ij=0;

for i=1:p

[zeta1, zeta2]= Polar;[eta1 ,eta2 ]=Polar;

a_ij=a_ij+(1/i)*(zeta1*(sqrt(2)*G2+eta2)...

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0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

t

X

X1

X2

X3

Figure 1: Numerical result of Example Run1 with Milstein approximation.

-zeta2*(sqrt(2)*G1+eta1));

end

I_ij=a_ij*Delta/(pi);

I_ij=I_ij+Delta*(G1*G2/2+sqrt(ro)*(mu1_j*G2-mu2_j*G1));

In Figure 1, we give the numerical result of Example Run1. We have thesame results as in [7].

Example Run 2. (Correlated Brownian motions) We recall the strongly-coupled Ornstein-Uhlenbeck process (14):

dX1

t = (11X1

t 12X2

t)dt+1dW1

t,dX2t = (21X1t 22X2t)dt+2dW2t,

where dW1tdW2t =dt and the transformed form (17) is

dX1t = (X1t 2X2t)dt+dZ1t,dX2t = (X1t X2t)dt+ 0.6dZ1t + 0.8dZ2t.

Our Taylor scheme with order 1/2 gives

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Y1n+1= Y1n + (Y2n 2Y1n ) +

1

2(3Y2n + 5Y

1n )

2 + Z1 2.6I10 0.8I20,

Y2n+1= Y2n + (Y2n Y1n ) +

1

2(2Y2n + 3Y

1n )

2 + 0.6Z1

+0.8Z2 1.6I10 0.8I20.

We compute the integrals I20, I10 as in the following lines:

%Approximation of I_j0 and I_0j

function [I_10,I_20]=ito_j0(p,Delta,G1,G2,mu1_j,mu2_j,ro)

a_10=0;a_20=0;for i=1:p

[eta1,eta2]=Polar;

a_10=a_10+(1/i)*eta1;

a_20=a_20+(1/i)*eta2;

end

I_10=a_10*(1/pi)*sqrt(Delta*2)+2*sqrt(Delta*ro)*mu1_j;

I_10=(1/2)*Delta*I_10+(1/2)*Delta*sqrt(Delta)*G1;

I_20=a_20*(1/pi)*sqrt(Delta*2)+2*sqrt(Delta*ro)*mu2_j;

I_20=(1/2)*Delta*I_20+(1/2)*Delta*sqrt(Delta)*G2;

The main file can be run as:

clf

randn(state,1)

T = 1; Delta = 2^(-9); delta = Delta^2;

L = T/Delta; K = Delta/delta;

X1 = zeros(1,L+1); X2 = zeros(1,L+1);

X1(1) = 1;X2(1) = 0.1;

p=2;ro=0;

for i=1:p

ro=ro+1/(i*i);ro=(pi*pi)/6-ro;ro=ro/(2*pi*pi);

endfor j = 1:L

G1 = randn; G2 = randn;

Winc2 = sqrt(Delta)*G2;Winc1 = sqrt(Delta)*G1;

[mu1, mu2 ]=Polar;

[I10,I20]=ito_j0(p,Delta,G1,G2,mu1,mu2,ro);

X1(j+1) = X1(j) +(-2*X1(j)-X2(j))*Delta +...

(0.5)*(3*X2(j)+5*X1(j))*Delta^2+Winc1-(2.6)*I10-(0.8)*I20;

X2(j+1) = X2(j) + (-X2(j)-X1(j))*Delta+(0.5)*(2*X2(j)...

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0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

ItoTaylor approximation for X1

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

Exact numerical simulation for X1

Figure 2: Comparison of exact numerical solution of Run2 with TaylorScheme of order 1/2 for X1.

+3*X1(j))*Delta^2+ (0.6)*Winc1 +(0.8)*Winc2-(1.6)*I10-(0.8)*I20;

end

plot([0:Delta:T],X1,r-), hold on

plot([0:Delta:T],X2,bl--)

xlabel(t,FontSize,16), ylabel(X,FontSize,16)

legend(X^1,X^2)

Exact solution of the system (14) is obtained in matrix formulation as:

Xt = X0exp(tA) + t0

exp((s t)A)BdZt, (21)

where A = 1 2

1 1

, and B = 1 0

0.6 0.8

.

We perform the exact numerical simulation for the system (14). We firstcompute the matrix multiplications in the Eqn. (21), and then approximatecomponentwise. In Figures 2-3, we compare the obtained results. It can beseen easily that they are almost the same.

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0 0.2 0.4 0.6 0.8 11.5

1

0.5

0

0.5

ItoTaylor approximation for X2

0 0.2 0.4 0.6 0.8 11.5

1

0.5

0

0.5

Exact numerical simulation for X2

Figure 3: Comparison of exact numerical solution of Run2 with TaylorScheme of order 1/2 for X2.

7 Conclusion and Outlook

This work have totally covered the numerical solutions of SDEs by means ofIto expansions. We obtained the stochastic Taylor series expansions of thesystems os SDEs with both correlated and independent Wiener processes.In the case of correlated processes, we first transformed the system into onehaving independent Brownian motions. And then, we obtained the Ito Tay-lor series expansions. Also, we performed the computations of the iteratedIto integrals with several Wiener processes by using MATLAB.

As a future work, the relation between the Ito Taylor approximations andstochastic control problems can be considered. Furthermore, as for ap-

plication to the more complicated systems of SDEs, e.g., Heston model,may be handled by means of Ito Taylor formula. A connection betweenthe iterated Ito integrals and Malliavin calculus should be investigated[1, 12, 13, 14, 15, 17, 19]. A detailed package for the systems of SDEs,which has a higher-order convergence, can be generated via MATLAB.

References

[1] Bell, D., 2007, The Malliavin Calculus, Dover.

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[2] Brummelhuis, R. Mathematical Methods Lecture Notes 6, Depart-

ment of Economics, Mathematics and Statistics, Birkbeck, Universityof London, 2009.

[3] Burrage, K. and Burrage, P.M., 1999, High strong order methods fornon-commutative stochastic ordinary differential equation system andthe Magnus formula, Physica D 133, 34-48.

[4] Burrage, K. and Burrage, P.M., 2000, Order conditions of stochasticRunge-Kutta methods by B-series, SIAM J. Numer. Anal. 38, 1626-1646.

[5] Burrage, P.M., 1999, Runge-Kutta Methods for Stochastic Differential

Equations, PhD Thesis, Department of Mathematics, University ofQueensland, Australia.

[6] Gilsing, H., and Shardlow, T., SDELab: A package for solving stochas-tic differential equations in MATLAB, Journal of Computational andApplied Mathematics 205 (2007), no. 2, 1002-1018.

[7] Higham, D. J., and Kloeden, P. E., MAPLE and MATLAB for Stochas-tic Differential Equations in Finance, 2002, in Programming Languagesand Systems in Computational Economics and Finance, 233-270.

[8] Hoffman, N., Platen, E., and Schweizer, M., 1992, Option pricing under

incompleteness and stochastic volatility, J. Mathematical Finance, 2,153-187.

[9] Ito K., 1944, Stochastic Integral, Proc. Imperial Acad. Tokyo, 20, 519-524

[10] Kloeden, P.E. and Platen, E., 1992, Numerical Solution of StochasticDifferential Equations, Springer-Verlag, Berlin.

[11] Kunita, H., 1990, Stochastic Flows and Stochastic Differential Equa-tions, Vol 24 of Cambridge Studies in Advanced Mathematics, Cam-bridge University Press.

[12] Kusuoka, S. and Stroock, D., 1982, Applications of Malliavin CalculusI, Stochastic Analysis, Proceedings Taniguchi International Sympo-sium Katata and Kyoto, 271-306.

[13] Kusuoka, S. and Stroock, D., 1985, Applications of Malliavin CalculusII, J. Faculty Sci. Uni. Tokyo Sect. 1A Math., 32, 1-76.

[14] Kusuoka, S. and Stroock, D., 1987, Applications of Malliavin CalculusIII, J. Faculty Sci. Univ. Tokyo Sect. 1A Math., 34, 391-442.

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[15] Malliavin, P., Thalmaier, A., 2005, Stochastic Calculus of Variations

in Mathematical Finance, Springer.

[16] Milstein, G.N., 1995, Numerical Integration of Stochastic DifferentialEquations, Kluwer, Dordrecht/Boston/London.

[17] Nualart, D., 2006, The Malliavin calculus and related topics (Secondedition ed.), Springer-Verlag.

[18] ksendal, B., 2000, Stochastic Differential Equations, An Introductionwith Applications, 5th edition

[19] ksendal, B., 1997, An Introduction To Malliavin Calculus With Ap-

plications To Economics, Thesis, Dept. of Mathematics, University ofOslo.

[20] Platen, E., 1999, An introduction to numerical methods for stochasticdifferential equations, Acta Numerica, 8, 197-246.

[21] Rumelin, W., 1982, Numerical treatment of stochastic differentialequations, SIAM J. Numer. Anal. 19, 604-613.

[22] Ylmaz, F., Oz, H., Weber, G.W., 2012, Change of Time Methodand Stochastic Taylor Expansion with Computation of Expectation,Book Chapter, Modeling, Optimization, Dynamics and Bioeconomy,

Springer, to appear.

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