Stochastic Calculus

An introduction

Jan Nygaard Nielsen

Department of Mathematical Modelling,Technical University of Denmark,

DK-2800 Lyngby - Denmark.jnn@imm.dtu.dk

Preface

This note contains a brief introduction to stochastic differential equations and their applicationin mathematical finance. It does not pretend to be complete in any respect. Thetopics of thisnote are covered in more detail in the course 04445Statistics in Finance, and the interestedreader may consult the adjacent lecture notes [8] or one of the standard texts [9, 4].

Throughout it is assumed that the reader is familiar with elementary probability theory andstatistics at a level corresponding to the DTU courses 01142Probability theory, 04041Intro-ductory Statisticsor 04040Introductory Statistics and Probability. No prior knowledge abouteconomics or finance is required, and the theory is formulated in the classical sense withoutreference to measure theory, albeit it is now common to introduce probability measure transfor-mations. The theory is much more conveniently formulated in terms of equivalent martingalemeasures and absolute continuous measure transformaations using Girsanov’s theorem, but thisis deemed to be outside the scope of this introductory note. See the references listed above fordetails.

The mathematics of modern finance theory utilizes some intimate relations between stochasticdifferential equations and (deterministic) partial differential equations. In particular, we will fo-cus on the Feynman-Kac representation theorems, which makes it possible solve some parabolicCauchy problems in terms of conditional expectations. These mathematical tools are essentialfor the pricing of financial derivatives, where only the simplest type of European options willbe considered here.

Any comments about misprints or suggestions for improvement will be mostly appreciated.

Jan Nygaard Nielsen

Contents

1 Stochastic Calculus 1

1.1 Dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Ito stochastic calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8

1.5 Stochastic differential equations . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.7 The Ito formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.8 Analytical solution methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.9 Feynman-Kac representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1 Stochastic Calculus

In this chapter, stochastic differential equations will be formally introduced. This exposition tostochastic calculus does not pretend to be complete. The reader is referred to[1, 6, 5, 10] for adetailed account.

1.1 Dynamical systems

Assume that we wish to model a general physical, chemical or technical system. Mathematicalmodelling of such systems often leads to the formulation of a system of coupled (nonlinear)differential equations, which may, in general, be written on the formdX(t)dt = _X(t) = f (t;X(t)); (1)

wheref(t;X(t)) describes the time-directed evolution of the socalledstate variablesX(t) 2Rn. The state variables describe the state of the system at timet in thestate space.

The derivation of these equations are often based on a number of conceptual, mathematical andnumerical approximations and the validity of these are difficult to evaluate perse.

By adding a stochastic term to (1) to account for these approximationsrandom differentialequationsare obtained as illustrated in these examples.

EXAMPLE 1.1 (MONEY MARKET ACCOUNT ). Consider a simple money market accountdB(t) = r(t)B(t)dt B(0) = 1 (2)

whereB(t) is the value of the account at timet, andr(t) denotes the interest rate.

It is very likely that the interest rate varies randomly in time, i.e.we haver(t) = ~r(t) + ”noise”

where~r(t) is assumed to be deterministic. If we insert this in (2), we getdB(t) = (~r(t) + �”noise”)B(t)dt B(0) = 1 (3)

where� denotes the standard deviation of the noise. The question is now how do we formalizethe concept of ”noise” such that (3) makes sense, and how do we solve it ? �EXAMPLE 1.2 (STOCK PRICES ). Empirical analysis shows that the uncertainty, or the so-called volatility, of stock prices, foreign exchange rates and interest rates depend on the currentlevel, i.e. dS(t) = �S(t)dt+ ”noise”S(t)dt S(0) = s (4)

which is essentially similar to (3). �

EXAMPLE 1.3 (SIMPLE BLACK -SCHOLES). If we consider a simple financial market withtwo assets:

1. A risky asset, where the price of the assetS(t) at timet is described by (4), and

2. A safe asset, namely the money market account (2).

we get the model dS(t) = �S(t)dt+ ”noise”S(t)dt S(0) = s (5a)dB(t) = r(t)B(t)dt B(0) = 1 (5b)

With one very important definition of the noise process in (5), namely the socalledBrownianmotion, (5) is the celebrated Black-Scholes model. This model will be described in great detaillater. �1.2 Preliminaries

In this section a number of basic concepts from probability theory will be introduced.

We assume the existence of a probability space(;F ;P), whereF is a�-algebra on the samplespace of possible outcomes,(;F) is a measurable space andP: F 7! [0; 1] is a probabilitymeasure.

DEFINITION 1.1 (A STOCHASTIC PROCESS). A stochastic processis a parametrized collec-tion of stochastic variables fX(s)gts=0 (6)

defined on a probability space(;F ;P) and assuming values inR. Note that for each fixedt 2 Rwe have a stochastic variable! ! X(t; !) : ! 2 (7)

On the other hand, fixing! 2 we can consider the functiont! X(t; !) : t 2 R (8)NIntuitively, it may be useful to think oft as “time“ and each! as an individual “particle“ oran “experiment“. ThusX(t; !) would represent the position (or result) at timet of the particle(experiment)!.

DEFINITION 1.2 (FILTRATION ). A filtration on (;F) is a family fFtg1t=0 of �-algebrasFt � F such that 0 � s < t ) Fs � Ft N2

Informally speaking,Fs denote the set of events (or the information set) up to times. Thenatural filtration fFtg1t=0 is increasingandright continuous, i.e. at timet, 0 � s < t, moreinformation is available (or, at least, information is not lost)Fs � Ft than at times and in thelimit complete information is obtainedF1 = F . Application of the natural filtrationfFtg1t=0implies that information aboutX(t) in (14) must be deduced from observations offX(s)gts=0.It is convenient to think of the natural filtration in terms of the flow of information such thatthe filtrationF1 contains all information about the past, present and future. A more reasonablefiltration isFt that contains information about the past and present, because it corresponds tothe concept of causality in system theory. The concept of filtration is important for computingconditional expectations and the following definition introduces another important conceptinthis context.

DEFINITION 1.3 (MARTINGALE ). A stochastic processfX(s)gts=0 defined on the probabilityspace(;F ;P) is called amartingalewith respect to a filtrationfFtg1t=0 if

1.X(t) is fFtg-measurable for allt2.EfjX(t)jg <1 for all t, and

3.EfX(t)jFsg = X(s) for all s � t NREMARK 1.1. Consider a stochastic variableX(t) as a function that maps the sample spaceintoR,X(t) : 7! R. If f! 2 : X(t)(!) � xg 2 F for eachx 2 R, thenX(t) is said to beFt-measurable. HDEFINITION 1.4 (ADAPTED PROCESS). The stochastic processX(t) is adaptedto the filtra-tionFt if, for eacht � 0,X(t) is anFt-measurable random variable. NREMARK 1.2 (ADAPTEDNESS). It is instructive to think of measurability and adaptedness inthe sense that if a functiong(t) is said to beFt-measurable, then it essentially means thatg(t)is known at timet. HA standard Wiener processis an abstract mathematical description of the physical process ofBrownian motion. The mathematical properties defining a Wiener process,fW (t); t � 0g, aregiven in

DEFINITION 1.5 (THE W IENER PROCESS). A stochastic processfW (t); t � 0g is said to bea Wiener processif it satisfies the following conditions

1.W (0) = 0 with probability 1 (w.p. 1)

2. The incrementsW (t1) �W (t0);W (t2) �W (t1); : : : ;W (tn) �W (tn�1) of the processfor any partitioning of the time interval0 � t0 < t1 < : : : < tn < 1 are mutuallyindependent

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3. The incrementsW (t)�W (s) for any0 � s < t are normally distributed with mean andvariance respectivelyEfW (t)�W (s)g = 0 (9)V fW (t)�W (s)g = jt� sj; (10)

i.e.W (t)�W (s) 2 N(0; jt� sj).4.W (t) has a continuous sample path

5. If a Wiener processW (t) is adapted to a given filtrationFt and possesses the propertythat W (t)�W (s) is independent ofFsthenW (t) is said to be aF -Wiener process. N

EXAMPLE 1.4. The Wiener process is a Markov process as well as a martingale. �Using this definition of the Wiener process, we shall now try to formalize the “noise“ processconsidered in Examples 1.1-1.3. The differentiability properties of the Wiener process have notbeen stated, so we will commence the discussion by considering the random difference equationX(t+�t)�X(t) = �(t;X(t))�t+ �(t;X(t))�W (t) (11)

where �W (t) =W (t+�t)�W (t) (12)

Thedrift function�(t;X(t)) accounts for the evolution of the mean in the small time interval�t > 0, whereas the squareddiffusion function�(t;X(t)) accounts for the evolution of thevariance.

It is obvious to try to obtain a random differential equation by dividing (11) through by�t andthen letting�t tend to 0. Formally we should obtain_X(t) = �(t;X(t)) + �(t;X(t))V (t) X(0) = x (13)

where we have included an initial valuex and introducedV (t) as the formal time derivative ofthe Wiener process.

Assuming thatV (t) is a well defined process, it should now be possible to solve (11) for ev-ery realization, trajectory or sample path ofV (t). It may be shown that the processV (t) isunfortunately not well defined as the Wiener process is nowhere differentiable, although it iscontinuous. For illustration consider the limitlimh!0 Ef(W (t+ h))2g �Ef(W (t))2gh = t+ h� th = 1

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Thus in a mean square sense the derivative of the Wiener processW (t) is not the derivativeprocessV (t) = _W (t) as defined above.

The sample paths (realizations) of the process is continuous with probability one, butthey arenowhere differentiable with probability 1 due to the (independent) increments, see eg. [10] for arigorous proof. Very informally speaking, it is simply too much to ask that a stochastic process,like the Wiener process, be both continuous and differentiable if there should be room for therandomness.

Another approach is to let�t tend to zero in (11) without dividing through by�t. Formally weget dX(t) = �(t;X(t))dt+ �(t;X(t))dW (t) X(0) = x (14)

and it is natural to interpret (14) as a shorthand notation for the following integral equationX(t) = x+ tZ0 �(s;X(s))ds+ tZ0 �(s;X(s))dW (s) (15)

Theds-integral may be interpreted as an ordinary Riemann-integral, whereas the natural inter-pretation of thedW (s)-integral is as an Riemann-Stieltjes integral for every trajectory ofW .Unfortunately this is not reasonable as it can be shown that the processW (t) is of unboundedvariation (this follows immediately from Definition 1.5), i.e. thedW (s)-integral in (15) is di-vergent.

Strictly speaking, the notation in (14) does not make any sense as it describes the infinitesimalevolution ofX(t), which is driven by a Wiener process with unbounded variation. We will,however, use the notation (14) for convenience repeatedly in the following, but itshould beremembered that it is only shorthand for (15).

The remaining questions are now� how do we formalize the stochastic integral in (15),� how do we define the adjacent stochastic calculus and� how do we analyze (14) in this framework.

1.3 Stochastic Integrals

Although the Wiener process has some simple probabilistic properties it is by no means simpleto define stochastic integration with respect to a Wiener process, because the trajectory of aWiener process is very odd. Let us list some of its peculiar properties� As a Wiener process is of unbounded variation, it will eventually hit every realvalue no

matter how large or how negative.� Once a Wiener process hits a value, it immediately hits it again infinitelyoften, and thenagain from time to time in the future.

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� It doesn’t matter what scale you examine a Wiener process on – it looks just the same.Thus a Wiener process or Brownian motion pertains the same selfsimilarityproperty asfractals.

Nevertheless, we intend to introduce the stochastic integraltZ0 g(s)dW (s); (16)

whereg(t) is some suitably smooth function by the following scheme, which is identical tothedefinition of the Riemann-integral

1. Partition the time interval[0; t] in n subintervals of equal length, i.e. define the timeinstants0 = t0 < t1 < : : : < tn = t.

2. Define for each trajectory! an approximate integralIn(!) byIn(!) = n�1Xk=0 g(�k; !)[W (tk+1; !)�W (tk; !)] (17)

where�k is some arbitrarily chosen time in the interval[tk; tk+1[.3. Finally, we letn tend to infinity and hope thatIn(!) tends to some limitI, which we will

use to define the integral (16).

As an example, let us consider the caseg(t) = W (t), i.e. we wish to compute the stochasticintegral tZ0 W (s)dW (s) (18)

where we choose to compute the integral fromt0 = 0 instead of the more generalt0, becausewe may use thatW (0) = 0 to obtain a shorter formula.

It is convenient to consider the quadratic variation ofW (t) on the interval[0; t], i.e. we com-mence by considering the integral tZ0 (dW )2Thus we introduce the notation�Wk = W (tk+1)�W (tk) and define the stochastic variableSn = n�1Xk=0(�Wk)2 (19)

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If the Wiener process was differentiable, we would expect thatSn would converge to zero asn tends to infinity. Let us introduce the subintervals�t = tk+1 � tk, i.e. �t = t=n. FromDefinition 1.5, it immediately follows thatEf(�Wk)2g = �tk and thusEfSng = n�1Xk=0 Ef(�Wk)2g = n�1Xk=0 �tk = tThe variance ofSn is found by direct calculationVfSng = n�1Xk=0 Vf(�Wk)2g = 2 n�1Xk=0(�tk)2 = 2n� tn�2 = 2t2nIn other words, we haveVfSng = Ef(Sn � EfSng)2g = Ef(Sn � t)2g = 2t2nand thus limn!1Ef(Sn � t)2g = 0In this case, we say thatSn converges towardst in a mean square sense or in the spaceL2. Thisunexpected result form the foundation of the socalledIto formula, which plays a fundamen-tal role in stochastic calculus as the stochastic counterpart of the well knownchain rule fromordinary calculus.

The main result may be restated as followstZ0 (dW )2 = tor in differential form

M ETATHEOREM 1. (dW )2 = dtFormally this metatheorem1 does not make any sense, but it worth noticing that it states that thesquare of a stochastic increment yields a purely deterministic property.

Let us return to the evaluation of (18). We proceed in a similar fashion as above by constructingsums of the form (19). We will consider two different sums which evaluates theW (t) part ateither the left hand side of the interval[tk; tk+1[, �k = tk, or the right hand side�k = tk+1, i.e.An = n�1Xk=0 W (tk)(W (tk+1)�W (tk)) (�k = tk) (20)Bn = n�1Xk=0 W (tk+1)(W (tk+1)�W (tk)) (�k = tk+1) (21)

1We will state a number of rules-of-thumb as metatheorems throughtout the remainder of these notes.

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We immediately get the identitiesAn +Bn = W 2(t) (22a)Bn �An = n�1Xk=0(�Wk)2 = Sn (22b)

for n!1, whereSn is given by (19). It immediately follows thatBn�An ! t, and inL2 weget limitsAn ! ABn ! Bwhere A = W 2(t)2 � t2 (23a)B = W 2(t)2 + t2 (23b)

These results show that the value of the stochastic integral (18) depends critically on the place-ment of�k in the interval[tk; tk+1[, i.e. the integral depends on where the integrand is evaluatedin the interval[tk; tk+1[. Needless to say, this is not the case in ordinary calculus.

By choosing�k = tk, we get the enormously importantIto-integral, which yieldstZ0 W (s)dW (s) = W 2(t)2 � t2By choosing�k = tk+1, we get tZ0 W (s)dW (s) = W 2(t)2 + t2Note that in both cases, we get the additional termt=2 compared to ordinary calculus.

There is a priori no particular reason for choosing to interpret stochastic calculus in the Ito sense,but it has some nice mathematical properties that will be discussed in the following section.

1.4 Ito stochastic calculus

In this section the Ito stochastic integral discussed above will be formally introduced.

EXAMPLE 1.5. As a simple example of these concepts, assume that we wish to compute theexpected value ofW (t)dW (t) with respect to the natural filtration, i.e.EfW (t)dW (t)jFtg

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As we use the natural filtration (the filtration generated by the Wiener process itself),W (t) isknown with respect toFt, i.e.EfW (t)dW (t)jFtg = W (t)EfdW (t)jFtgFurthermore, we know that the incrementdW (t) of the Wiener process is independent of thepast, so we may consider the unconditional expected valueW (t)EfdW (t)jFtg = W (t)EfdW (t)gFinally, as the mean of the incrementdW (t) is zero, cf. Definition 1.5, we get the resultEfW (t)dW (t)jFtg = 0We will later show that this result is valid for a large class ofFt-measurable functionsg(t), i.e.Efg(t)dW (t)jFtg = 0However, this result only applies for stochastic integration in the Ito sense. �DEFINITION 1.6 (THE CLASS L2 OF SQUARE–INTEGRABLE FUNCTIONS ). LetL2[a; b]de-note the class of processesg that satisfies the conditionsg isFt � adapted (24a)bZa Ef(g(s))2gds <1 (24b)NWe will now for somea � b define the stochastic integralbZa g(s)dW (s) (25)

for all g 2 L2[a; b]. We will only consider simple functions (to be defined below) and leave thegeneralization to the interested reader.

Assume thatg is simple, i.e. there exists deterministic time instantsa = t0 < t1 < : : : < tn = bsuch that g(s) = g(tk) for s 2 [tk; tk+1[where g(tk) 2 Ftk k = 0; : : : ; nIn other wordsg(tk) isFtk-measurable, i.e.g(tk) is known at timetk.

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For a simple processg we define the stochastic integral by a sum similar to (20), i.e.bZa g(s)dW (s) = n�1Xk=0 g(tk)(W (tk+1)�W (tk)) (26)

It is inherently important that we define the incremental Wiener process in terms of the forwarddifferencesW (tk+1) �W (tk).THEOREM 1.1 (STOCHASTIC I NTEGRATION RULES ). Let g andh be simple processes thatsatisfies (24) and let�; � be real numbers. The following rules applyE8<: bZa g(s)dW (s)9=; = 0 (27)E8><>:0@ bZa g(s)dW (s)1A29>=>; = bZa Efg2(s)gds (Ito isometry) (28)E8<:0@ bZa g(s)dW (s)1A0@ bZa h(s)dW (s)1A9=; = bZa Efg(s)h(s)gds (29)bZa g(s)dW (s) isFb �measurable (30)E8<: bZa g(s)dW (s)jFs9=; = 0 (31)bZa (�g(s) + �h(s))dW (s) = � bZa g(s)dW (s) + � nZa h(s)dW (s) (32)

Proof. For easy of notation, we introduce the entitiesgk = g(tk); �Wk = W (tk+1)�W (tk); �tk = tk+1 � tk; Fk = FtkWe get E8<: bZa g(s)dW (s)9=; = n�1Xk=0 Efgk�Wkg

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If we use the fact that the processg is adapted, we getEfgk�Wkg = EfEfgk�WkjFkgg = EfgkEf�WkjFkgg;where we have used the standard trick to introduce a conditioning argument and taken expec-tation with respect to that argument. As the Wiener process have independent increments, weget EfgkEf�WkjFkgg = 0and we have proved (27).

Next we will prove (28). By introducing the well known sum, we getE8><>:0@ bZa g(s)dW (s)1A29>=>; =Xi;j Efgigj(�Wi)(�Wj)gwhere we need to consider two cases

1. Fori = j, we getEfg2i (�Wi)2g = EfEfg2i (�Wi)2jFigg= Efg2iEf(�Wi)2ggjFigg = Efg2i�tig= Efg2i g�t2. Fori 6= j with, sayi < j, we getEfgigj(�Wi)(�Wj)g = EfEfgigj(�Wi)(�Wj)jFjgg= Efgigj(�Wi)Ef(�Wj)jFjgg = 0

as the Wiener increment has the conditional mean 0.

Thus we have E8><>:0@ bZa g(s)dW (s)1A29>=>; =Xi;j Efg2i g�t = bZa Efg2i (s)gdsEq. (29) may be shown in a similar fashion. Eq. (30) follows immediately from the definitionof the stochastic integral, and (31) is shown as (27). The last rule is trivial and the proof is leftas an exercise for the reader. �REMARK 1.3 (ITO ISOMETRY ). Note that (28) establishes an isometry between stochasticintegrals and deterministic integrals. This is very useful for computing the variance of thesolution of a SDE. H

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REMARK 1.4. The rules in Theorem 1.1 may be extended to cover a larger class of functionsthan the simple functions considered above by considering Cauchy-sequences of simple func-tions, but we will not go into the details here. HREMARK 1.5. It is possible to extend stochastic integration to all adapted processesg thatsatisfies the condition

P

8<: tZ0 g2(s)ds <19=; = 1For all suchg it is not guaranteed that (27), (28) and (31) are valid, but the properties (30) and(32) are valid. HIt is easy to show that the Wiener process is in itself aF -martingale and it is an very importantconsequence of Theorem 1.1 that the martingale property is preserved with respect to integrationof L2-processes.

THEOREM 1.2 (CONTINUOUS TRAJECTORIES ). Assume thatg 2 L2[0; t] for all t � 0.Define the processX by X(t) = tZ0 g(s)dW (s) (33)

ThenX(t) is a martingale with continuous trajectories.

Proof. By direct calculation we getX(t) = tZ0 g(u)dW (u) = sZ0 g(u)dW (u) + tZs g(u)dW (u)= Xs + tZs g(u)dW (u)Using (27) we get EfX(t)jFsg = Xs + E8<: tZs g(u)dW (u)jFs9=; = XsThe continuity of the trajectories is difficult to prove, but it should be intuitively clear as theWiener process lacks jumps. �

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1.5 Stochastic differential equations

In the following exposition to stochastic differential equations, we will onlyuse the Wienerprocess as the driving noise process. We recall that the Wiener process is both a Markov processand a martingale, and that the mean of the stochastic integral (in the Ito sense) of any squareintegrable, adapted process with respect to a Wiener process is zero.

Stochastic differential equations driven by eg. a Poisson process (orjump processes, countingprocessesor marked point processes) are gaining ground in the financial literature. However,a considerable extension of the measure-theoretical concepts of adaptedness and predictabilityare required, which is beyond the scope of this lecture note. It is duly noted that the topicscovered in this chapter may be generalized to cover the very general class ofsquare integrableprocesses, see eg. [2, 6, 5] for details.

We repeat that the notion of stochastic differential equations (SDEs) is merely a shorthandnotation for stochastic integrals. The latter may be defined in several ways, but we restrictour discussion to stochastic integrals in the Ito sense. Unfortunately, thisimplies that the wellknown chain rule for variable transformations must be replaced by the socalledIto formula,which will be introduced in the multivariate case. This formula may be used to obtain closedform solutions of some SDEs. Beside that, it just makes Ito stochastic calculus more tedious.

We assume the existence of a probability space(;F ;P), whereF is a�-algebra on the samplespace of possible outcomes,(;F) is a measurable space andP: F 7! [0; 1] is a probabilitymeasure. Let the drift� : R 7! R and the diffusion� : R 7! R be Borel-measurable functions2

and assume thatXt : 7! R is a solution to the time-homogeneous Ito stochastic differentialequation dX(t) = �(X(t))dt+ �(X(t))dWt; X(0) = x0 (34)

wherefW (t); t � 0g is a standard Wiener process defined on the probability space(;F ;P)equipped with the natural filtrationfFtg generated byW (t).The standard Wiener process is defined in Definition 1.5, the concepts of filtration,martingalesand adaptedness are defined in Definitions 1.2, 1.3 and 1.4.

Let us give a number of examples to motivate the following discussion.

EXAMPLE 1.6 (THE W IENER PROCESS). Consider the Wiener processdX(t) = �dW (t); X(0) = x0 (35)

where� is the standard deviation of the process andx0 is a deterministic initial condition, whichis short for X(t) = x0 + tZ0 �dW (s)From the definition of the Wiener process (Definition 1.5), it immediately follows thatX(t) = x0 + �(W (t)�W (0)) = x0 + �W (t)

2The functions� and� will, in general, depend on ap-dimensional parameter vector� 2 � � Rp, where�may be some constrained subset ofRp. For notational convenience this parameter dependency will be suppressedin this chapter.

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Next we compute the mean ofX(t), i.e.EfX(t)g = E8<:x0 + tZ0 �dW (s)9=; = x0which follows from (27). The variance is given byVfX(t)g = V8<:x0 + tZ0 �dW (s)9=; = �2 E8<:0@ tZ0 dW (s)1A29=;= �2 tZ0 Ef12gds = �2t;where we have used the Ito isometry property (28). This shows thatVfX(t)g !1 ast!1.However the process is still bounded in finite time. �EXAMPLE 1.7 (WIENER PROCESS WITH DRIFT ). Let us compute the mean and variance ofX(t), whereX(t) is the solution todX(t) = �dt+ �dW (t); X(0) = x0where� and� are some constants. This SDE corresponds toX(t) = x0 + tZ0 �ds + tZ0 �dW (s)As in the previous example, we getEfX(t)g = x0 + E8<: tZ0 �ds9=; + E8<: tZ0 �dW (s)9=; = x0 + �tVfX(t)g = V8<: tZ0 �dW (s)9=; = �2 E8<:0@ tZ0 dW (s)1A29=; = �2tWe see that the mean ofX(t) has a linear trend (or drift). �EXAMPLE 1.8 (STOCHASTIC EXPONENTIAL GROWTH ). Consider the SDEdX(t) = �X(t)dt+ �dW (t); X(0) = x0 (36)

where� and� are constants, which may describe unlimited growth in biological systems or astochastic money market account.

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If we take expectations in the adjacent stochastic integral equation, we getEfX(t)g = x0 + E8<: tZ0 �X(s)ds9=; + E8<:� tZ0 dW (s)9=;Of course, the last term equals zero. Using Fubini’s theorem (which we will neither state norprove here), we may exchange the expectation and integration operators, i.e.EfX(t)g = x0 + E8<: tZ0 �X(s)ds9=; = x0 + � tZ0 E fX(s)g dsCompared to the last two examples the problem is now thatEfX(t)g exists on both sides of theequation. A standard trick is to introducem(t) = EfX(t)g and then take the expectation andderivative with respect to timet on both sides. i.e.dm(t)dt = _m(t) = �m(t); m(0) = EfX(0)gwhich clearly has the solutionEfX(t)g = m(t) = m(0) exp(�t)We see thatEfX(t)g grows exponentially ast!1. �Considering the slightly more complicated, yet familiarGeometric Brownian Motion (GBM)dX(t) = �X(t)dt+ �X(t)dW (t) (37)

where� and� are positive constants, it is not clear if there is existence and uniqueness ofthe solution for allt � 0 or if the solution might blow up with positive probability in finitetime. Along the same lines we must examine whether it is possible to determinea closed formsolution or not. In the former case, we may have to impose some restrictions on the functions�and� in (34) in order to obtain existence of the solution.

It is an interesting result that the answers to these questions only depends on the properties ofthe infinitesimal characteristics� and� in (34) (and possibly the initial conditionX(0)).1.6 Existence and uniqueness

As for ordinary differential equations (ODEs) Lipschitz and bounded growth conditionsmustbe imposed on the drift and diffusion terms in order to obtain existence and uniquenessofsolutions.

We must distinguish betweenweakandstrongsolutions to (34). A strong solution is obtainedif the driving Wiener process is given in advance as a part of the problem such that the obtainedsolution to (34) isFt-adapted, whereFt is the�-algebra generated by the Wiener process. Onthe other hand, if we are just given the infinitesimal characteristics� and� in advance and the

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solution should apply for all possible Wiener processes, then the obtained solution iscalled aweak solution. It is clear that a strong solution is also a weak solution, because the particularWiener processW (t) that resulted in the strong solution is just one of infinitely many Wienerprocesses that will give a weak solution. The converse is not true in general.

THEOREM 1.3 (STRONG UNIQUENESS). Suppose that the infinitesimal characteristics�(x)and�(x) are locally Lipschitz-continuous in the state variable; i.e. for every integern � 1 thereexists a constantCn such that for everyt � 0, jxj � n andjyj � n:j�(x)� �(y)j+ j�(x)� �(y)j � Cnjx� yj (38)

Then strong uniqueness holds for (34).

Proof. Omitted. See [6]. �Let us consider an example that does not satisfy the condition (38).

EXAMPLE 1.9. It is easy to verify that the differential equation_x(t) = 3x2=3has several solutions, for anya > 0,x(t) = (0 for t � a(t� a)3 for t > aThis ODE is excluded as�(x) = 3x2=3 does not satisfy (38) forx = 0. �We will need an additional assumption in order to obtain existence and uniqueness of solutionsof (34).

ASSUMPTION 1.1 (LINEAR GROWTH ). The functions� and� satisfy the usual linear growthcondition j�(x)j+ j�(x)j � K(1 + jxj); 8x 2 R (39)

whereK is a positive, real constant. NEXAMPLE 1.10.The differential equation_x(t) = x2(t); x(0) = 1corresponding to�(x) = x2 has the solutionx(t) = 11� t ; 0 � t < 1Thus it is impossible to find a solution for allt. This is due to the fact that�(x) = x2 does notsatisfy Assumption 1.1. �

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Next consider an example of a SDE.

EXAMPLE 1.11 (TRESPASSING IN A MINEFIELD ). Consider as an example of a process thatsatisfies (38), but not (39)dX(t) = �12 exp(�2X(t))dt+ exp(�X(t))dW (t)ForX(t) < 0, we get exponential growth, which is faster than linear growth, and (39) is notsatisfied. It may be shown that the solution is given byX(t) = ln[W (t) + exp(X(0))]It is clearly seen that forW (t) < � exp(X(0)), we should compute the natural logarithm of anegative number! If we define the timeT (X(0); !) byT (X(0); !) = minft � 0 :W (t; !) = � exp(X(0; !))g; ! 2 it is clear that the solution only exists up to timeT (X(0); !). This explosion timedepends onthe stochastic initial condition and the actual trajectory of the driving Wienerprocess. �EXAMPLE 1.12 (GEOMETRIC BROWNIAN M OTION ). Consider the Geometric Brownian Mo-tion given in (37). In this case an explosion timee may be defined bye = infft : X(t) =2 (0;1)gwhich states that the explosion timee is the first (i.e. smallest) time, where the processX(t)hits the boundary 0 or takes the value of1. Note that it is also critical ifX(t) attains the value0 because the processX(s) will remain at zero fors � t. The value ofX(t) ast!1 dependson the parameters� and� as follows (this is illustrated in Example 1.14):

1. If � > 12�2 thenX(t)!1 ast!1 a.s.

2. If � < 12�2 thenX(t)! 0 ast!1 a.s.

3. If � = 12�2 thenX(t) will fluctuate between arbitrary large and arbitrary small values ast!1 a.s.

It may, however, be shown thatX(t) does not take either the value 0 or1 in finite time.Hence the Geometric Brownian Motion does not explode. This is also clear as the infinitesimalcharacteristics are linear inX(t) and thus fulfills the Lipschitz condtions (38) and, in particular,the linear growth condition (39). �For one-dimensional processes (34), the asssumptions (38) and (39) are not necessary toensurenonexplosive solutions. In the remainder of this lecture note (and the problems), we will simplyassume that a unique solution exists. For brevity we will not, in general, list therestrictions onthe parameters that must be imposed to ensure nonexplosiveness.

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1.7 The Ito formula

An important feature of Ito stochastic differential equations is stated in

THEOREM 1.4 (THE I TO FORMULA ). Let X(t) be a solution to (34) and' : R 7! R be aC1;2(R)-function applied toX(t) Y (t) = '(t;X(t)) (40)

Then the following chain rule appliesdY (t) = �@'@t + � @'@X(t) + 12�2 @2'@X(t)2� dt+ � @'@X(t)dW (t) (41)

where the functions� and� are as defined prior to (34).

Proof. For notational brevity, we will leave out the argument in'(t;X(t));X(t) andW (t) inthis ad hoc proof. A second order Taylor expansion ofd' givesd' = @'@t dt+ @'@xdX + 12 @2'@x2 (dX)2 + 12 @2'@t2 (dt)2 + 12 @2'@t@xdtdXFrom (34), we get (dX)2 = �2(dt)2 + �2(dW )2 + 2��(dt)(dW )Compared to terms withdt anddW , the terms containing(dt)2 and(dt)(dW ) are insignificant.Thus we getd' = @'@t dt+ @'@x (�dt+ �dW ) + 12�2@2'@x2 (dW )2= @'@t dt+ �@'@xdt+ �@'@xdW + 12�2@2'@x2 dt= �@'@t + �@'@x + 12�2@2'@x2 � dt+ �@'@xdWwhere we have also used Metatheorem 1. �REMARK 1.6 (SHORT FORM OF THE I TO FORMULA ). By introducing the notation't = @'@tetc. (41) may be written asd' = ('t + �'x + 12�2'xx)dt+ �'xdW; (42)

where eg.'t should not be confused with'(t). HREMARK 1.7 (ADDITIONAL TERM IN THE I TO FORMULA ). As opposed to classical calcu-lus (41) contains the additional term12�2 @2'(Xt)@X2t , which makes Ito calculus more complicatedfor theoretical considerations, although solutions to (34) are both Markov processesand Mar-tingales. H

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REMARK 1.8. It follows from the last remark that the diffusion term from (34) enters the driftof (41). Another remarkable observation from (41) is that the transformed variableY (t) is alsodescribed by an Ito diffusion process. HEXAMPLE 1.13.Consider the integralI = Z t0 W (s)dW (s)ChooseX(t) = W (t), which implies thatdX(t) = dW (t), ie. � = 0 and� = 1. In additionchoose the transformation'(t; x) = 12x2. ThenY (t) = '(t;W (t)) = 12W (t)2Using (41), we getdY (t) = @'@t dt+ @'@xdW (t) + 12 @2'@x2 (dW (t))2= 0 +W (t) � dW (t) + 12(dW (t))2= W (t) � dW (t) + 12 � dtThis implies that d(12(W (t))2) = W (t) � dW (t) + 12dtor in integral form 12(W (t))2 = Z t0 W (s)dW (s) + 12tor I = Z t0 W (s)dW (s) = 12(W (t))2 � 12t �EXAMPLE 1.14 (GEOMETRIC BROWNIAN M OTION ). We wish to solve the SDE given bydX(t) = �X(t)dt+ �X(t)dWt; X0 > 0 (43)

This SDE is called the Geometric Brownian motion and is considered extensively in mathe-matical finance as a model for interest rates and stock prices. This is mainly due to the factthat the solutionX(t) is log-normally distributed and thus excludes negative interest rates (orpopulations in biology or concentrations in chemistry).

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By introducing the transformationY (t) = '(t;X(t)) = ln(X(t)), we get@'@t = 0; @'@X(t) = 1X(t) ; @2'@X(t)2 = � 1X(t)2Inserting these in (41) yieldsdYt = ��X(t) 1X(t) + 12�2X(t)2�� 1X(t)2�� dt+ �X(t) 1X(t)dW (t)or d(lnX(t)) = (�� 12�2)t+ �dW (t)and, finally, X(t) = X0 exp((�� 12�2)t+ �W (t)); (44)

It may be shown that the log-stock returns are normally distributed, i.e.ln(X(t)=X0) 2 N((�� 12�2)t; �2t) (45)�1.8 Analytical solution methods

Generally, it is difficult to obtain closed form solutions to stochastic differential equations,but the Ito formula, that in all other aspects complicates analytical calculations considerably,may be valuable as an intermediary step in obtaining closed form solutions to (34). Someexamples along these lines will be given. As with linear ordinary differential equations, thegeneral solution of a linear stochastic differential equation can be found explicitly.

Closed form solutions for a number of SDEs (linear and nonlinear) are listed in [7], where avery elaborate discussion of numerical solutions may be found as well.

The general form of aunivariate linear stochastic differential equationisdX(t) = (�1(t)X(t) + �2(t))dt+ (�1(t)X(t) + �2(t))dW (t); X(t0) = X0 (46)

where the coefficients�1; �2; �1 and�2 are given functions of timet or constants. We assumethat these functions are measurable and bounded on an interval0 � t � T such that theexistence and uniqueness theorem from the preceding section applies and ensures the existenceof a strong solutionX(t) on t0 � t � T for each0 � t0 < T .

When all the functions are constants the SDE is said to beautonomousand its solutions arehomogeneous Markov processes. Otherwise, the SDE is said to benonautonomous. When�2(t) � 0 and�2(t) � 0, (46) reduces to thehomogenouslinear SDEdX(t) = �1(t)X(t)dt+ �1(t)X(t)dW (t); X(t0) = X0 (47)

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which clearly has the solutionX(t) � 0. The socalledfundamental solution�t;t0 which satisfiesthe initial condition�t0;t0 = 1 is much more important as all other solutions may be expressedin terms of the fundamental solution. To determine�t;t0, we consider the simple case where�1(t) � 0, i.e. dX(t) = (�1(t)X(t) + �2(t))dt+ �2(t)dW (t); X(t0) = X0 (48)

where the Wiener process appears additively. In this case we say that the SDE is linear in thenarrow-sense.

THEOREM 1.5 (SOLUTION TO A LINEAR SDE IN THE NARROW -SENSE). The solution to(48) is given byX(t) = �t;t00@Xt0 + tZt0 �2(s)��1s;t0ds + tZt0 �2(s)��1s;t0dW (s)1A (49)

where �t;t0 = exp0@ tZt0 �1(s)ds1A (50)

Proof. The homogenous version (�2(t) � 0) of (48) is an ordinary differential equation_X(t) = �1(t)X(t) (51)

with the fundamental solution �t;t0 = exp0@ tZt0 �1(s)ds1AApplying the Ito formula (41) to the transformation'(t; x) = ��1t;t0x and the solutionX(t) of(51), we getd(��1t;t0) = d��1t;t0dt X(t) + (�1(t)X(t) + �2(t))��1t;t0! dt+ �2(t)��1t;t0dW (t)= �2(s)��1t;t0dt+ �2(t)��1t;t0dW (t) (52)

as d��1t;t0dt = ���1t;t0�1(t)The right hand side of (52) can be integrated to give��1t;t0X(t) = ��1t;t0X(t0) + tZt0 �2(s)��1s;t0ds+ tZt0 �2(s)��1s;t0dW (s)Since��1t0;t0 = 1, we have obtained the solution (49). �

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REMARK 1.9. Notice that��1t;t0 means1=�t;t0 and not the inverse function. HTHEOREM 1.6 (SOLUTION TO A LINEAR SDE IN THE WIDE -SENSE). The solution to (46)is given byX(t) = �t;t00@Xt0 + tZt0 (�2(s)� �1(s)�2(s))��1s;t0ds + tZt0 �2(s)��1s;t0dW (s)1A (53)

where�t;t0 is given by (50).

Proof. Omitted. See [7][Section 4.3]. �THEOREM 1.7 (MOMENTS OF A LINEAR SDE IN THE WIDE -SENSE). The meanm(t) =EfX(t)g of (46) satisfies the ordinary differential equation_m(t) = �1(t)m(t) + �2(t); m(0) = m0 (54)

and the second order momentP (t) = EfX2(t)g satisfies_P (t) = (2�1(t) + �21(t))P (t) + 2m(t)(�2(t) + �1(t)�2(t)) + �22(t); P (0) = P (55)

Proof. By proceeding as in Example 1.8 (54) is readily seen. In order to show (55), we applythe Ito formula (41) to the transformation'(t; x) = x2, i.e.d' = �0 + (�1X + �2)2X + 12(�1X + �2)2 � 2� dt+ (�1X + �2)2XdW= �2(�1X2 + �2X) + �21X2 + �22 + 2�1�2X� dt+ 2(�1X2 + �2X)dWwhere the arguments have been left out for brevity as in the following equivalent stochasticintegral formulationX2 = X20 + tZt0 �2(�1X2 + �2X) + �21X2 + �22 + 2�1�2X� ds+ tZt0 2(�1X2 + �2X)dWBy taking expectations the last term drops out, cf. (27). If we defineP (t) = EfX2(t)g and takederivatives, we obtain_P (t) = 2�1(t)P (t) + 2�2(t)m(t) + �21(t)P (t) + �22(t) + 2�1(t)�2(t)m(t)which equals (55). �REMARK 1.10.Recall that the varianceVfX(t)g may be determined fromVfX(t)g = P (t)� (m(t))2 (56)H

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In order to solve (55) the following result from calculus may be useful.

REMARK 1.11 (THE ”A RMOUR ” FORMULA ). The solution to the ODE_x(t) + (t)x(t) = #(t); t 2 I (57)

where ; # : I ! R are continuous in the intervalI, is given byx(t) = exp(�(t))�Z exp((t))#(t)dt+ c� ; t 2 I; c 2 R (58)

where (t) = Z (t)dt HAs an example consider the SDE from Example 1.8 again.

EXAMPLE 1.15.Consider theLangevin equationdX(t) = ��X(t)dt+ �dW (t); X(0) = X0 (59)

Without loss of generality, we assume thatt0 = 0. From (50), we immediately get�t;0 = exp0@� tZ0 �ds1A = exp(��t)and thus (49) yields the solutionX(t) = exp(��t)0@X0 + � tZ0 exp(�s)dW (s)1Awhich is called theOrnstein-Uhlenbeck process.

The meanm(t) = EfX(t)g is obtained from (54), i.e.m(t) = m0 exp(��t)The second momentP (t) should fulfill_P (t) + 2�P = �2Using Remark 1.11, we get (t) = tZ0 2�dt = 2�t

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and insertion in (58) yieldsX(t) = exp(�2�t)0@ tZ0 exp(2�s)�2ds+ P1A= P exp(�2�t) + �22� (1 � exp(�2�t))The variance may be found as stated in Remark 1.10, i.e.VfX(t)g = P exp(�2�t) + �22� (1� exp(�2�t))�m20 exp(�2�t)and the stationary value is limt!1VfX(t)g = �22�Note that it is not just�2. �1.9 Feynman-Kac representation

In this section we will describe a close relationship between stochasticdifferential equationsand parabolic partial differential equations (PDEs).

Consider the followingCauchy problem@F@t (t; x) + �(t; x)@F@x (t; x) + 12�2(t; x)@2F@x2 (t; x) = 0 (60a)F (T;X) = �(X) (60b)

where the functions�(t; x), �(t; x) and�(T; x) are given, and we wish to determine the func-tionF (t; x).As opposed to solving (60) analytically, we will consider a representation formula for the solu-tionF (t; x) in terms of an associated stochastic differential equation.

Assume that there exists a solution to (60). Fix the timet and the statex. Let the stochasticprocessX(t) be a solution to the SDEdX(s) = �(s;X(s))ds+ �(s;X(s))dW (s); X(t) = x (61)

wheres is now the running time.

REMARK 1.12 (SAME �(�) AND �(�)). Note that the functions�(t;X(t)) and�(t;X(t)) in (60)and (61) are the same - except for the fact that the running time variable in (61) iss. HIf we apply the Ito formula (41) to the processF (s;X(s)) and write the result in stochastic

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integral form, we getF (T;X(T )) = F (t;X(t))+ TZt �@F@t (s;X(s)) + �(s;X(s))@F@x (s;X(s))+ 12�2(s;X(s))@2F@x2 (s;X(s))� ds+ TZt �(s;X(s))@F@x (s;X(s))dW (s) (62)

Let us further assume that the process�(s;X(s))@F@x (s;X(s))belongs to the spaceL2[t; T ], see Definition 1.6. If we use thatF (t; x) solves (60), then theds-integral drops out of (62). If we apply the boundary conditionF (T; x) = �(x), the initialconditionX(t) = x and take the expected value of the remaining parts of (62) then the last termalso drops out, cf. (27). The only remaining term isF (t; x) = Et;xf�(X(T ))g (63)

where the subscriptt; x on the expectation operator is used to emphasize the fixed initial condi-tionX(t) = x.

We state this important result in a theorem.

THEOREM 1.8 (THE FEYNMAN -K AC REPRESENTATION 1). Assume thatF solves the bound-ary problem (60) and that the process�(s;X(s))@F@x (s;X(s)) 2 L2 for t � T; x 2 R (64)

whereX(t) is defined by (61). ThenF has the stochasticFeynman-Kac-representationF (t; x) = Et;xf�(X(T ))g (65)

Proof. Follows from the preceding derivation. �Note that the theorem simply states that the solution to (60) is obtained as the expected value ofthe boundary condition.

REMARK 1.13.A major problem with this approach is that it is impossible to check the as-sumption (64) in advance as it requires a priori information about the solutionF to do so. Atleast two things can go wrong

1. Eq. (60) does not have a ”sufficiently integrable” solution, i.e. the process (64) does notbelong to the classL2. If the latter is the case, the solution offered by the Feynman-Kacrepresentation is pure nonsense.

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2. The solution to (60) is not unique. If there are more solutions, the Feynman-Kac approachjust supplies the ”sufficiently integrable” solution. The remaining solutions must befoundin another way.

In this lecture note, we will assume that all the functions in question are ”sufficiently integrable”.We will not go into all the technical details, see eg. [2, 10]. HLet us consider a couple of examples of this remarkable approach.

EXAMPLE 1.16.We wish to solve the following boundary problem in the domain[0; T ]�R:@F@t + �x@F@x + 12�2x2@2F@x2 =F (T; x) = ln(x2)where� and� are assumed to be constants.

It is readily seen that the associated SDE is given bydX(s) = �X(s)ds + �X(s)dW (s); X(t) = xWe recognize this as the Geometric Brownian Motion from Example 1.14 on page 19, wherethe solution was found to beX(T ) = exp�ln(x) + (�� 12�2)(T � t) + �[W (T )�W (t)]�Using Theorem 1.8, we get the resultF (t; x) = Et;xf2 ln(X(T ))g= 2 ln(x) + 2(�� 12�2)(T � t)as the expected value of the Wiener incrementW (T )�W (t) is zero. �We will now consider a more general case.

THEOREM 1.9 (THE FEYNMAN -K AC REPRESENTATION 2). Let the functions�, � and�be given as above, and letr be a constant. The solution to@F@t (t; x) + �(t; x)@F@x (t; x) + 12�2(t; x)@2F@x2 (t; x) + rF (t; x) = 0F (T; x) = �(x)is given by F (t; x) = exp(r(T � t)) Et;xf�(X(T ))g (66)

where the processX(t) is given by (61).

Proof. Omitted. See eg. [2]. �26

1.10 Notes

Some of the material in this chapter is sampled from the excellent lecture note [2], who alsoextends the theory to cover general jump processes and provides financial applications. A morethorough treatment is given by eg. [1, 7, 10]. In particular the monograph [7] covers alargenumber of interesting topics – also of practical interest. The often referenced books by [6, 5, 3]are also recommended, although they require some understanding of measure theory.

References

[1] Arnold, L. Stochastic differential equations, John Wiley & Sons, New York. 1974.

[2] Bjork, T. Stokastisk Kalkyl och Kapitalmarknadsteori, Matematiska Institutionen, KTH,Stockholm. 1994.

[3] Doob, J.L.Stochastic Processes, John Wiley & Sons, New York. 1990.

[4] Duffie, D. Dynamic Asset Pricing Theory, Second Edition, Princeton University Press,Princeton, New Jersey. 1996.

[5] Ikeda, N. and Watanabe, S.Stochastic Differential Equations and Diffusion Processes,North Holland/Kodansha, Amsterdam. 1989.

[6] Karatzas, I. and Shreve, S. E.Brownian Motion and Stochastic Calculus, Second Edition,Springer-Verlag, New York. 1996

[7] Kloeden, P.E. and Platen, E.Numerical Solutions of Stochastic Differential Equations, Sec-ond Edition, Springer-Verlag, Heidelberg. 1995.

[8] Madsen, H. and Nielsen, J. N. and Baadsgaard, M.Statistics in Finance - Lecture notes,Department of Mathematical Modelling, Lyngby, Denmark. 1997.

[9] Musiela, M. and Rutkowski, M.Martingale Methods in Financial Modelling, Springer.1997.

[10] Øksendal, B.Stochastic Differential Equations, 4th Edition, Springer-Verlag, Heidelberg.1995.

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