Stochastic Analysis and its Applications1

Hiroshi Toyoizumi 2

March 8, 2013

1This handout can be found at www.f.waseda.jp/toyoizumi.2E-mail: toyoizumi@waseda.jp

Contents

1 Introduction 5

2 Example: Stochastic Differential Equation 72.1 Ordinary Differential Equation . . . . . . . . . . . . . . . . . . . 72.2 Stochastic Differential Equation . . . . . . . . . . . . . . . . . . 7

3 Example: Solving Elliptic Equation 93.1 Elliptic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Solving Dirichlet Problem using Stochastic Calculus . . . . . . . 9

4 Probability Rules and Random Variables 114.1 Definition of Probability . . . . . . . . . . . . . . . . . . . . . . 114.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Normal Distribution and Geometric Brownian Motion 135.1 What is Normal Random Variable? . . . . . . . . . . . . . . . . . 135.2 Lognormal Random Variables . . . . . . . . . . . . . . . . . . . 145.3 Lognormal Random Variables as the Stock Price Distribution . . . 165.4 Geometric Brownian Motions . . . . . . . . . . . . . . . . . . . 185.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Idea of Ito Calculus 226.1 Differentiability Issue . . . . . . . . . . . . . . . . . . . . . . . . 226.2 Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 236.3 Ito-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

7 Stochastic Integral 257.1 Diffusion Process . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2

CONTENTS 3

7.2 Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.3 Definition of Stochastic Integral . . . . . . . . . . . . . . . . . . 287.4 Martingale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.5 Calculus of Stochastic Integral . . . . . . . . . . . . . . . . . . . 30

8 Examples of Stochastic Integral 338.1 Evaluation of E[W (t)4] . . . . . . . . . . . . . . . . . . . . . . . 338.2 Evaluation of E[eαW (t)] . . . . . . . . . . . . . . . . . . . . . . . 35

9 Differential Equations 369.1 Ordinary Differential Equation . . . . . . . . . . . . . . . . . . . 369.2 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . 379.3 Stochastic Process and Partial Differential Equation . . . . . . . . 38

10 Martingale Representation of Random Variables 4010.1 Why the Representation Theorem? . . . . . . . . . . . . . . . . . 4010.2 Intuitive Explanation . . . . . . . . . . . . . . . . . . . . . . . . 4110.3 Constructing Martingale Representation . . . . . . . . . . . . . . 41

11 Simple Image of Malliavin Calculus 4311.1 Differential Calculus of Power Series . . . . . . . . . . . . . . . . 4311.2 Differential Calculus based on Integral by Parts . . . . . . . . . . 4411.3 Differential Calculus for Random Variables . . . . . . . . . . . . 44

12 Iterated Ito-integral and Chaos Expansion 4512.1 Iterated Ito-integral and Hermite polynomials . . . . . . . . . . . 4512.2 Chaos Exapnasion . . . . . . . . . . . . . . . . . . . . . . . . . . 4612.3 Skorohod integral . . . . . . . . . . . . . . . . . . . . . . . . . . 4612.4 Malliavin Derivative . . . . . . . . . . . . . . . . . . . . . . . . 4712.5 Clarck-Ocone Formula . . . . . . . . . . . . . . . . . . . . . . . 47

Bibliography

T. Bjork. Arbitrage Theory in Continuous Time. Oxford Finance. Oxford Univ Pr,2nd edition, 2004.

G. Di Nunno, B. Oksendal, and F. Proske. Malliavin calculus for Levy processeswith Applications to Finance. Springer Verlag, 2009.

R. Durrett. Probability: Theory and Examples. Thomson Learning, 1991.

G. Folland. Real analysis: modern techniques and their applications. Wiley NewYork, 1984.

B. Oksendal. Stochastic Differential Equations: An Introduction With Applica-tions. Springer-Verlag, 2003.

H. Toyoizumi. Performance Evaluation based on Stochastic Analysis. 2009. URLhttp://www.f.waseda.jp/toyoizumi/.

4

Chapter 1

Introduction

Here’s the schedule of this class:

1. Introduction

2. Examples of Stochastic Calculus 1

3. Examples of Stochastic Calculus 2

4. Random Variables and Other Probability Rules

5. Brownian Motions

6. Simple Image of Ito-integral

7. Construction of Ito-integral 1

8. Construction of Ito-integral 2

9. Martingale Representation Theorem

10. Stochastic Differential Equation

11. Application of Ito integral to Finance Problems

12. Simple Image of Malliavin Calculus

13. Iterated Ito-integral

14. Winer Chaos Expansion

5

6 CHAPTER 1. INTRODUCTION

15. Clark-Ocone Representation Theorem

Here’s text book for this class:

• Bernt Oksendal, Stochastic differential equations: an introduction with ap-plications, Springer-Verlag New York, Inc., 2003 Oksendal [2003]

• Giulia Di Nunno, Bernt Oksendal,Frank Proske, Malliavin Calculus forLevy Processes with Applications to Finance, Springer-Verlag New York,Inc., 2008 Di Nunno et al. [2009]

Chapter 2

Example: Stochastic DifferentialEquation

Give examples to understand why we need stochastic analysis.

2.1 Ordinary Differential EquationProblem 2.1. Solve an ordinary differential equation;

dxdt

= ax, (2.1)

with an appropriate initial condition.

Problem 2.2. What do you think is the main contribution in solving (9.1).

Problem 2.3. Consider examples where (9.1) is useful in the fields such as busi-ness, finance, biology, physics, chemistry and so on.

2.2 Stochastic Differential EquationSuppose we need to incorporate some randomness in (9.1). Here’s one option:

dXdt

= (a+W )X , (2.2)

where W represents some randomness.

7

8 CHAPTER 2. EXAMPLE: STOCHASTIC DIFFERENTIAL EQUATION

Problem 2.4. Solve (2.2) formerly.

Definition 2.1 (white noise?). The White noise W =W (t) may be defined as thedifferential of Brownian motion B,

W =dBdt

. (2.3)

Alternately, the white noise can be the limit of binomial process. Let ∆ be asmall interval and n = t/∆. Define a random walk on the discrete time i∆ suchas

S(i∆)−S((i−1)∆) =

1√n with probability 1/2,

− 1√n with probability 1/2,

(2.4)

for all n. Then formerly,

S→ B. (2.5)

Problem 2.5. So, S→ B. What is W .

Problem 2.6. What is the white noise? What kind of features does it supposed tohave? Do those features affect the solution?

The differential equation (2.2) cannot be solved as the same manner with (9.1).Actually, the expression of (2.2) is not mathematically correct. In order to incorpo-rate randomness properly into (9.1), we have to introduce Ito-stochastic calculus!

Chapter 3

Example: Solving Elliptic Equation

Give another example to understand why we need stochastic analysis.

3.1 Elliptic EquationProblem 3.1. Solve an ordinary differential equation;

d2 fdx2 = 0, (3.1)

with a boundary condition f (0) = 0 and f (1) = 1.

Definition 3.1 (Dirichlet problm). More generally, consider a domain U in Rn.Given a function f on the boundary of U . Find a function f satisfying a differentialequation;

∆ f = 0, (3.2)

with an appropriate initial condition. These kind of problems are called Dirichletproblmes.

Problem 3.2. Consider a real life example of Dirichlet problem.

3.2 Solving Dirichlet Problem using Stochastic Cal-culus

Dirichlet problem can be solved by using Brownian motion B as

f (x) = E[ f (Bτ)|B(0) = x], (3.3)

9

10 CHAPTER 3. EXAMPLE: SOLVING ELLIPTIC EQUATION

where τ is the hitting time to the boundary of U .

Problem 3.3. Draw the picture of the above Brownian motion. Describe an intu-itive reasoning behind this arguement.

Chapter 4

Probability Rules and RandomVariables

We check the basic properties needed to use rigorous stochastic analysis. Morecasual treatment of probability can be find Toyoizumi [2009].

4.1 Definition of ProbabilityDefinition 4.1 (σ -algebra). A family of subset on Ω, F is called σ -algebra, if

1. /0 ∈F

2. F ∈F → Fc ∈F

3. A1,A2, · · · ∈F →∪∞i=1Ai ∈F

Example 4.1 (Borel σ -algebra). The smallest σ -algebra U contains all open sub-set is called Borel σ -algebra.

Definition 4.2 (Probability). Consider a function P on a measurable space (Ω,F )satisfies:

1. P : F → [0,1]

2. P( /0) = 0,P(Ω) = 1

3. A1,A2, · · · ∈F :disjoint

→ P(∪∞i=1Ai) =

∞

∑i=1

P(Ai) (4.1)

11

12 CHAPTER 4. PROBABILITY RULES AND RANDOM VARIABLES

Problem 4.1. What is σ -algebra, F ? Why do we need σ -algebra in the definitionof probability? Consider what will happen when we didn’t use σ -algebra.

Definition 4.3. We say an event F occurs almost surely, if P(F) = 1.

Problem 4.2. Why do we say “almost sure” instead of saying “for sure”?

Problem 4.3. Consider an example that an event A with P(A) = 0 but A maysomehow occur (or is not prohibited to occur).

Problem 4.4. Try Exercise 2.3 in Oksendal [2003].

For further detail of measure theory, which is useful to understand probability,you may check Folland [1984] for example.

4.2 Random VariablesDefinition 4.4 (F -measurable). A function Y : Ω→R is called F -measurable if

Y−1(U) := ω ∈Ω;Y (ω) ∈U ∈F , (4.2)

for all open sets U ∈ R.

Definition 4.5 (random variable). A F -measurable function X on a probabilityspace (P,Ω,F ) is called a random variable.

Problem 4.5. So, what are these random variables? Why do we call them vari-ables instead of functions?

Definition 4.6 (Distribution of X).

µX(B) = P(X−1(B)), (4.3)

for B ∈F .

Problem 4.6. Does this definition of distribution of X contradict with the ordinarydefinition of the distribution function of random variable X?

Problem 4.7. Show

E[X ] :=∫

Ω

X(ω)dP(ω) =∫

RxdµX(x) (4.4)

Chapter 5

Normal Distribution and GeometricBrownian Motion

5.1 What is Normal Random Variable?Let’s begin with the definition of normal random variables.

Definition 5.1 (Normal random variable). Let X be a random variable with itsprobability density function

dPX ≤ x= f (x)dx =1

(2π)1/2σe−(x−µ)2/2σ2

dx, (5.1)

for some µ and σ . This is called the normal random variable with the parametersµ and σ , or written by N[µ,σ2].

Problem 5.1. Draw the graph of the probability density function f (x) of normalrandom variable N[µ,σ ].

Theorem 5.1 (Mean and variance of normal random variables). Let X be a normalrandom variable with the parameters µ and σ . Then, we have the mean

E[X ] = µ, (5.2)

and the variance

Var[X ] = σ2. (5.3)

Proof. omit.

13

14CHAPTER 5. NORMAL DISTRIBUTION AND GEOMETRIC BROWNIAN MOTION

Definition 5.2 (Standard normal random variable). Let X be a normal randomvariable with µ = 0 and σ = 1. The random variable is called the standard normalrandom variable, which is often denoted by N[0,1].

Lemma 5.1. Let X be a normal random variable with its mean µ and standarddeviation σ . Set Z =(X−µ)/σ . Then, Z is the standard normal random variable.

Proof. See Exercise 5.5.

Theorem 5.2. Let (Xi)i=1,2,...,n are independent normal random variables withits mean µi and standard deviation σi. Then the sum of these random variablesX = ∑

ni=1 Xi is again a normal random variable with

µ =n

∑i=1

µi, (5.4)

σ2 =

n

∑i=1

σ2i . (5.5)

Proof. We just prove X satisfies (5.4) and (5.5). We have

µ = E[X ] = E

[n

∑i=1

Xi

]=

n

∑i=1

E [Xi] =n

∑i=1

µi.

Also, we have

σ2 = E[X ] =Var

[n

∑i=1

Xi

]=

n

∑i=1

Var [Xi] =n

∑i=1

σ2i .

We can prove that X is a normal random variable by using so-called charac-teristic function method (or Fourier transform).

So, it is very comfortable to be in the world of normal random variables. Veryclosed!

5.2 Lognormal Random VariablesNow it is the turn of Lognormal random variable. Lognormal random variablesare used to model the stock price distribution in Mathematical Finance.

5.2. LOGNORMAL RANDOM VARIABLES 15

Definition 5.3 (lognormal random variable). The random variable Y is said to belognormal if log(Y ) is a normal random variable.

Thus, a lognormal random variable can be expressed as

Y = eX , (5.6)

where X is a normal random variable. Lognormal random variables plays a mea-sure role in finance theory!

Theorem 5.3 (lognormal). If X is a normal random variable having the mean µ

and the standard deviation σ2, the lognormal random variable Y = eX has themean and the variance as

E[Y ] = eµ+σ2/2, (5.7)

Var[Y ] = e2µ+2σ2− e2µ+σ2

. (5.8)

Remark 5.1. It is important to see that although the mean of the lognormal randomvariable is subjected to not only the mean of the original normal random variablebut also the standard deviation.

Problem 5.2. Explain why the mean E[Y ] is larger than the “expected” meaneE[X ].

Proof. Let us assume X is the standard normal random variable, for a while. Letm(t) be the moment generation function of X , i.e.,

m(t) = E[etX ]. (5.9)

Then, by differentiate the left hand side and setting t = 0, we have

m′(0) =ddt

m(t)|t=0 = E[XetX ]|t=0 = E[X ]. (5.10)

Further, we have

m′′(0) = E[X2].

Problem 5.3. Derive m′′(0) = E[X2].

16CHAPTER 5. NORMAL DISTRIBUTION AND GEOMETRIC BROWNIAN MOTION

On the other hand, since X is the standard normal random variable, we have

m(t) = E[etX ] =1√2π

∫∞

−∞

etxe−x2/2dx.

Since tx− x2/2 = t2− (x− t)2/2, we have

m(t) =1√2π

et2/2∫

∞

−∞

e−(x−t)2/2dx,

where the integrand of the right hand side is nothing but the density of the normalrandom variable N(t,1). Thus,

m(t) = et2/2.

More generally, when X is a normal random variable with N(µ,σ), we can obtain

m(t) = E[etX ] = eµt+σ2t2/2. (5.11)

(See exercise 5.6.) Since Y = eX , we have

E[Y ] = m(1) = eµ+σ2/2, (5.12)

and

E[Y 2] = m(2) = e2µ+2σ2. (5.13)

Thus,

Var[Y ] = E[Y 2]−E[Y ]2 = e2µ+2σ2− e2µ+σ2

. (5.14)

5.3 Lognormal Random Variables as the Stock PriceDistribution

Let S(n) be the price of a stock at time n. Let Y (n) be the growth rate of the stock,i.e.,

Y (n) =S(n)

S(n−1)(5.15)

5.3. LOGNORMAL RANDOM VARIABLES AS THE STOCK PRICE DISTRIBUTION17

In mathematical finance, it is commonly assumed that Y (n) is independent andidentically distributed as the lognormal random variable. Taking log on both sideof (5.15), then we have

logS(n) = logS(n−1)+X(n), (5.16)

where if X(n) = logY (n) is regarded as the error term and normally distributed,the above assumption is validated.

Example 5.1 (Stock price rises in two weeks in a row). Suppose Y (n) is thegrowth rate of a stock at the n-th week, which is independent and lognormallydistributed with the parameters µ and σ . We will find the probability that thestock price rises in two weeks in a row.

First, we will estimate Pthe stock rises. Since it is equivalent that y > 1 andlogy > 0, we have

Pthe stock rises= PS(1)> S(0)

= P

S(1)S(0)

> 1

= P

log(

S(1)S(0)

)> 0

= PX > 0

where X = logY (1) is N(µ,σ2), and we can define

Z =X−µ

σ, (5.17)

as the standard normal distribution. Hence, we have

PS(1)> S(0)= P

X−µ

σ>

0−µ

σ

= PZ >−µ/σ= PZ < µ/σ,

where we used the symmetry of the normal distribution (see exercise 5.1).Now we consider the probability of two consecutive stock price rise. Since

Y (n) is assumed to be independent, we have

18CHAPTER 5. NORMAL DISTRIBUTION AND GEOMETRIC BROWNIAN MOTION

Pthe stock rises two week in a row= PY (1)> 1,Y (2)> 1= PY (1)> 0PY (2)> 0= PZ < µ/σ2

Problem 5.4. 1. Derive the probability that stock price down two days in arow.

2. Derive the probability that stock price up at the first and down at the secondday.

5.4 Geometric Brownian MotionsDefinition 5.4 (Geometric Brownian motion). We say S(t) is a geometric Brow-nian motion with the drift parameter µ and the volatility parameter σ if for anyy≥ 0,

1. the growth rate S(t + y)/S(t) is independent of all history of S up to t, and

2. log(S(t + y)/S(t)) is a normal random varialble with its mean µy and itsvariance σ2y.

Let S(t) be the price of a stock at time t. In mathematical finance theory, often,we assume S(t) to be a geometric Brownian motion. If the price of stock S(t) is ageometric Brownian motion, we can say that

• the future price growth rate is independent of the past price, and

• the distribution of the growth rate is distributed as the lognormal with theparameter µt and σ2t..

The future price is probabilistically decided by the present price. Sometimes,this kind of stochastic processes are refereed to a Markov process.

Lemma 5.2. If S(t) is a geometric Brownian motion, we have

E[S(t)|S(0) = s0] = s0et(µ+σ2/2). (5.18)

5.4. GEOMETRIC BROWNIAN MOTIONS 19

Proof. Set

Y =S(t)S(0)

=S(t)s0

.

Then, Y is lognormal with (tµ, tσ2). Thus by using Theorem 5.3 we have

E[Y ] = et(µ+σ2/2).

On the other hand, we have

E[Y ] =E[S(t)]

s0.

Hence, we have (5.18).

Remark 5.2. Here the mean of S(t) increase exponentially with the rate µ +σ2/2,not the rate of µ . The parameter σ represents the fluctuation, but since the log-normal distribution has some bias, σ affects the average exponential growth rate.

Theorem 5.4. Let S(t) be a geometric Brownian motion with its drift µ andvolatility σ , then for a fixed t ≥ 0, S(t) can be represented as

S(t) = S(0)eW , (5.19)

where W is the normal random variable N(µt,σ2t).

Proof. We need to check that (5.19) satisfies the second part of Definition 5.4,which is easy by taking log on both sides in (5.19) and by seeing

log(

S(t)S(0)

)=W. (5.20)

Sometimes instead of Definition 5.4, we use the following stochastic differen-tial equation to define geometric Brownian motions,

dS(t) = µS(t)dt +σS(t)dB(t), (5.21)

where B(t) is the standard Brownian motion and dB(t) is define by so-called Itocalculus.

20CHAPTER 5. NORMAL DISTRIBUTION AND GEOMETRIC BROWNIAN MOTION

Note that the standard Brownian motion is continuous function but nowheredifferentiable, so the terms like dB(t)/dt should be treated appropriately. Thus,the following parts is not mathematically rigorous. The solution to this equationis

S(t) = S(0)e(µ−σ2/2)t+σB(t). (5.22)

To see this is the solution, we need “unordinary” calculus. We will see the detailin Theorem 9.2.

Problem 5.5. Use “ordinary” calculus to obtain the derivative,

dS(t)dt

, (5.23)

formally. Check if it satisfies (5.21), or not.

5.5 ExercisesExercise 5.1. Prove the symmetry of the normal distribution. That is, let X be thenormal distribution with the mean µ and the standard deviation σ , then for any x,we have

PX >−x= PX < x. (5.24)

Exercise 5.2 (moment generating function). Let X be a random variable. Thefunction m(t) = E[etX ] is said to be the moment generating function of X .

1. Prove E[X ] = m′(0).

2. Prove E[Xn] = dn

dtn m(t)|t=0.

3. Rewrite the variance of X using m(t).

Exercise 5.3. Let X be a normal random variable with the parameters µ = 0 andσ = 1.

1. Find the moment generating function of X .

2. By differentiation, find the mean and variance of X .

5.5. EXERCISES 21

Exercise 5.4. Use Microsoft Excel to draw the graph of probability distributionf (x) = dPX ≤ x/dx of normal random variable X with

1. µ = 5 and σ = 1,

2. µ = 5 and σ = 2,

3. µ = 4 and σ = 3.

What can you say about these graphs, especially large x? Click help in your Excelto find the appropriate function. Of course, it won’t help you to find the answerthough...

Exercise 5.5. Let X be a normal random variable with its mean µ and standarddeviation σ . Set Z = (X−µ)/σ . Prove Z is the standard normal random variable.

Exercise 5.6. Verify (5.11) by using Lemma 5.1

Exercise 5.7. Let Y = eX be a lognormal random variable where X is N(µ,σ).Find E[Y ] and Var[Y ].

Chapter 6

Idea of Ito Calculus

We extend the ordinary integral to stochastic integral based on brownian motions,which is quite different with normal (Riemann) integral.

6.1 Differentiability IssueConsider a differential equation(2.2);

dXdt

= (a+W )X , (6.1)

where

W =dBdt

(6.2)

and B is the standard Brownian motion. Brownian motion is continuous but un-fortunately nowhere differentiable. Indeed,

Theorem 6.1 (Differentiability of Brownian Motion). Brownian motions are Holdercontinuous with exponent γ < 1/2. However, with probability 1, they are not Lip-schitz continuous at any point

Proof. See for example, Durrett [1991].

Apparently, (6.1) won’t work.

Problem 6.1. So, how can we proceed to model the process of X? Consider atleast two options we may use.

22

6.2. INTEGRAL EQUATION 23

6.2 Integral EquationOne option is to use integral equation instead of differential equation of the type:

dXdt

=

(a+

dBdt

)X . (6.3)

Problem 6.2. Derive an appropriate integral equation.

Problem 6.3. What are the advantages of integral equations over differentialequations?

6.3 Ito-integralLet B be a standard Brownian motion. It is important to consider an integral:∫ t

0BdB. (6.4)

Problem 6.4. Solve (6.4) as usual. Take the expectation.

Approximate (6.4) as the sum of the rectangles of the form:

N

∑i=1

B(ti)∆B(ti) =N

∑i=1

B(ti)(B(ti+1)−B(ti)). (6.5)

Problem 6.5. Take the expectation on (6.5). What is happening here?

On the other hand, we have a nice formula:

∆B(ti)2 = B(ti+1)2−B(ti)2 = (∆B(ti))2 +2B(ti)∆B(ti). (6.6)

Thus,

B(t)2 =N

∑i=1

(∆B(ti))2 +2N

∑i=1

B(ti)∆B(ti) (6.7)

Problem 6.6. Explain why the first term (so-called quadratic variation) will con-verges to t.

24 CHAPTER 6. IDEA OF ITO CALCULUS

Thus, we have the Ito-integral:∫ t

0BdB =

12(B2

t − t). (6.8)

Problem 6.7. Explain why we have −t/2.

Problem 6.8. Compare the integral with

N

∑i=1

B(ti+1)(B(ti+1)−B(ti)), (6.9)

N

∑i=1

B((ti+1 + ti)/2)(B(ti+1)−B(ti)) (6.10)

Chapter 7

Stochastic Integral

7.1 Diffusion ProcessDiffusion process is a building block of stochastic integral.

Definition 7.1 (Diffusion process). A stochastic process X(t) is said to be a dif-fusion process if

X(t +∆t)−X(t) = µ(t,X(t))∆t +σ(t,X(t))Z(t). (7.1)

Here, Z(t) is independent normally distributed disturbance term, µ is drift term,and σ is so-called diffusion term.

Definition 7.2 (Wiener process). A stochastic process W (t) is said to be a Wienerprocess if

1. W (0) = 0.

2. Independent increment, i.e., for r < s≤ t < u

W (u)−W (t),W (s)−W (r) (7.2)

are independent.

3. For s < t, W (t)−W (s) is normally distributed as N[0, t− s].

4. W(t) has a continuous trajectory.

Remark 7.1. Sometimes, Wiener process is called by Brownian motion.

25

26 CHAPTER 7. STOCHASTIC INTEGRAL

Using the Wiener process W (t) in (7.1), we can rewrite it as

X(t +∆t)−X(t) = µ(t,X(t))∆t +σ(t,X(t))∆W (t), (7.3)

where ∆W (t) =W (t+∆t)−W (t). Dividing both sides with ∆t and letting ∆t→ 0,we have

dX(t)dt

= µ +σv(t). (7.4)

Here,

v(t) =dW (t)

dt. (7.5)

Remark 7.2. The process v(t) cannot be exist, while the Wiener process W (t)exists mathematically. Indeed W (t) cannot be differentiable almost everywhere.

Instead of dividing both sides with ∆t, just take ∆t→ 0 in (7.1), then we havedX(t) = µdt +σdW (t),X(0) = a.

(7.6)

Integrating (7.28) over [0, t], we have

X(t) = a+∫ t

0µds+

∫ t

0σdW (s). (7.7)

The term∫ t

0 µds is normal integral, so it is OK. However, we have a problem forthe term

∫ t0 σdW (s), which cannot be defined by ordinary integral.

Thus, in the following section, we need to study the integral like∫ t

0g(s)dW (s). (7.8)

7.2 InformationDefinition 7.3 (Information of X(t)). Define F X

t by the information generated byX(s) on s ∈ [0, t]. Roughly speaking, F X

t is “what has happened to X(s) up totime t.”

7.2. INFORMATION 27

When we can decide an event A happened or not by the the trajectory of X(t),we say that A is F X

t -measurable, and

A ∈F Xt . (7.9)

When a random variable Z can be completely determined by the trajectory of X(t),we write

Z ∈F Xt . (7.10)

Definition 7.4. A stochastic Y is said to be adopted to the filtration F Xt when

Y (t) ∈F Xt , (7.11)

for all t.

Example 7.1. These are examples of information.

1.

A = X(s)≤ 3.14 for all s≤ 9. (7.12)

Then, A ∈F X9 .

2.

A = X(10)> 8. (7.13)

Then, A ∈F X10, but A 6∈F X

9 .

3.

Z =∫ 5

0X(s)ds. (7.14)

Then, Z ∈F X5 .

4. Let W be the Wiener process, and

X(t) = sups≤t

W (s). (7.15)

Then, X is adopted to F Xt . However, when

Y (t) = sups≤t+1

W (s), (7.16)

Y is not adopted to F Xt .

28 CHAPTER 7. STOCHASTIC INTEGRAL

7.3 Definition of Stochastic IntegralDefinition 7.5.

g ∈L 2[a,b], (7.17)

when

1.∫ b

a E[g(s)2]ds < ∞,

2. g is adopted FWt .

Our objective is to define the stochastic integral∫ b

a g(s)dW (s) on g∈L 2[a,b].Assume g is simple, i.e.,

g(s) = g(tk) for s ∈ [tk, tk+1). (7.18)

Then, the stochastic integral can be defined by

∫ b

ag(s)dW (s) =

n−1

∑k=0

g(tk)[W (tk+1−W (tk)]. (7.19)

Remark 7.3. Note that this stochastic integral is forward incremental.

Problem 7.1. Prove g ∈L 2[a,b].

When g is not simple, we will proceed as follows:

1. Approximate g with simple functions gn.

2. Take n→ ∞ in ∫ b

agn(s)dW (s)→

∫ b

ag(s)dW (s). (7.20)

Theorem 7.1. Here are some important properties for our stochastic integral.When g ∈L 2, we have

1.

E[∫ b

ag(s)dW (s)

]= 0, (7.21)

7.4. MARTINGALE 29

2.

E

[(∫ b

ag(s)dW (s)

)2]=∫ b

aE[g2(s)]ds, (7.22)

3. ∫ b

ag(s)dW (s):F X

b -measurable. (7.23)

Proof. We prove the case only when g(t) is simple. Set g(t) = A1[a,b] with A ∈F X

a . Since A is independent with the future W (b)−W (a), we have

E[∫ b

ag(s)dW (s)

]= E[A(W (b)−W (a))]

= E[A]E[W (b)−W (a)] = 0.

Similarly, we have

E

[(∫ b

ag(s)dW (s)

)2]= E[A2(W (b)−W (a))2]

= E[A2]E[(W (b)−W (a))2]

= E[A2]t

=∫ b

aE[g2(s)]ds.

We can prove the general case using the approximation with simple functions.

7.4 MartingaleDefinition 7.6. Given a filtration Ft (the history observed up time t), a stochasticprocess Xt is said to be martingale, when

1. Xt is adopted to Ft ,

2. E[|X(t)|] for all t,

30 CHAPTER 7. STOCHASTIC INTEGRAL

3. for all s≤ t,

E[X(t)|Ft ] = X(s). (7.24)

Theorem 7.2. Every stochastic integral is martingale. More precisely,

X(t) =∫ t

0g(s)dW (s), (7.25)

is an FWt -martingale.

Theorem 7.3. A stochastic process X(t) is martingale if and only if

dX(t) = g(t)dW (t), (7.26)

which means X(t) has no dt-term.

7.5 Calculus of Stochastic IntegralWe consider a stochastic process X(t) satisfying following stochastic integral:

X(t) = a+∫ t

0µ(s)ds+

∫ t

0σ(s)dW (s). (7.27)

Equivalently, we write dX(t) = µ(t)dt +σ(t)dW (t),X(0) = a.

(7.28)

(7.28) is called stochastic differential equation.

Remark 7.4. We can see the infinitesimal dynamics more easily in the stochasticdifferential equations.

Remark 7.5. Wiener process W (t) is continuous but perfectly rugged. Thus, weneed unusual Ito-calculus.

Since W (t)−W (s) is N[0, t− s],

E[∆W ] = 0, (7.29)

E[(∆W )2] = ∆t, (7.30)Var[∆W ] = ∆t, (7.31)

Var[(∆W )2] = 2(∆t)2. (7.32)

7.5. CALCULUS OF STOCHASTIC INTEGRAL 31

For small ∆t, Var[(∆W )2] rapidly vanish, so deterministically we have

(∆W )2 ∼ ∆t. (7.33)

Thus, we can set

(dW )2 = dt, (7.34)

or ∫ t

0(dW )2 = t. (7.35)

Remark 7.6. Intuitive proof can be found in text Bjork [2004] p26.

Theorem 7.4 (Ito’s formula). Given a stochastic process X(t) with

dX(t) = µdt +σdW (t), (7.36)

set

Z(t) = f (t,X(t)). (7.37)

Then, we have the following change of variables rule:

dZ(t) = d f (t,X(t)) (7.38)

=

∂ f∂ t

+µ∂ f∂x

+12

σ2 ∂ 2 f

∂x2

dt +σ

∂ f∂x

dW (t), (7.39)

or more conveniently,

d f =∂ f∂ t

dt +∂ f∂x

dX +12

∂ 2 f∂x2 (dX)2. (7.40)

Roughly speaking, Ito’s formula reveals that the dynamics of a function ofstochastic process depends not only on the first order but also the second order ofthe underlying stochastic process.

Proof. Take Taylor expansion of f ,

d f =∂ f∂ t

dt +∂ f∂x

dX +12

∂ 2 f∂x2 (dX)2 +

12

∂ 2 f∂ t2 (dt)2 +

∂ 2 f∂x∂ t

dtdX . (7.41)

32 CHAPTER 7. STOCHASTIC INTEGRAL

Note that the the term (dX)2 should be ignored in normal differential, but instochastic differential this terms survive as we see in the following.

By (7.36), formally,

(dX)2 = µ2(dt)2 +2µσdtdW +σ

2(dW )2. (7.42)

Now we can igonre those terms with (dt)2 and dtdW which converges much fasterthan dt. Also, by (7.34), (dW )2 = dt. Substituting these, we have the result.

Lemma 7.1. You can use the following conventions:

(dt)2 = 0, (7.43)dtdW = 0, (7.44)

(dW )2 = dt. (7.45)

Theorem 7.5 (n-dimensional Ito’s formula). Given an n-dimensional stochasticprocess

X(t) = (X1(t),X2(t), . . . ,Xn(t)), (7.46)

with n-dimensional diffusion equation,

dX(t) = µdt +σdW (t), (7.47)

set

Z(t) = f (t,X(t)). (7.48)

Then, we have the following change of variables rule:

d f =∂ f∂ t

dt +∑i

∂ f∂xi

dXi(t)+12 ∑

i, j

∂ 2 f∂xix j

dXi(t)dX j(t). (7.49)

Corollary 7.1. Let X(t) and Y (t) be diffusion processes, then

d(X(t)Y (t)) = X(t)dY (t)+Y (t)dX(t)+dX(t)dY (t). (7.50)

Proof. Use Ito’s formula (Theorem 7.5) and

dWi(t)dWj(t) = δi jdt, (7.51)dWidt = 0. (7.52)

Chapter 8

Examples of Stochastic Integral

For pricing via arbitrage theory, we need to evaluate the expectation of Wienerprocess. We can use Ito’s formula for the evaluation.

8.1 Evaluation of E[W (t)4]

We already know that the mean and variance of W (t) by definition, i.e.,

E[W (t)] = 0, (8.1)Var[W (t)] = t. (8.2)

But how about the higher moments?

Theorem 8.1. Let W (t) be a Wiener process, then

E[W (t)4] = 3t2. (8.3)

Proof. Let f (t,x) = x4 and

Z(t) = f (t,W (t)) =W (t)4. (8.4)

In Ito’s formula (Theorem 7.4), we can set µ = 0 and σ = 1 and,

dX(t) = 0dt +1dW (t) = dW (t). (8.5)

Then,

dZ(t) =∂ f∂ t

dt +∂ f∂x

dX +12

∂ 2 f∂x2 (dX)2. (8.6)

33

34 CHAPTER 8. EXAMPLES OF STOCHASTIC INTEGRAL

Check∂ f∂ t

= 0, (8.7)

∂ f∂x

= 4x3, (8.8)

∂ 2 f∂x2 = 12x2, (8.9)

∂ 2 f∂ t2 = 0, (8.10)

and we have

dZ(t) = 6W (t)2dW (t)2 +4W (t)3dW (t). (8.11)

Since dW (t)2 = dt by Lemma 7.1, we can rewrite the dynamics for Z(t) =W (t)4

as d(W (t)4) = 6W (t)2dt +4W (t)3dW (t),Z(0) = 0.

(8.12)

Integrating the both side of (8.12), we have

W (t)4 = 6∫ t

0W (s)2ds+4

∫ t

0W (s)3dW (s). (8.13)

Taking the expectation, we have

E[W (t)4] = 6E[∫ t

0W (s)2ds

]+4E

[∫ t

0W (s)3dW (s)

](8.14)

= 6∫ t

0E[W (s)2]ds+4E

[∫ t

0W (s)3dW (s)

]. (8.15)

Since W (s) is N[0,√

s], we have

E[W (s)2]= s. (8.16)

Also, by Theorem ??,

E[∫ t

0W (s)3dW (s)

]= 0. (8.17)

Note that we used the fact W (s)3 ∈L 2. Now we can evaluate (8.14).

E[W (t)4] = 6∫ t

0sds = 3t2. (8.18)

8.2. EVALUATION OF E[EαW (T )] 35

8.2 Evaluation of E[eαW (t)]

Theorem 8.2. Let W (t) be a Wiener process, then

E[eαW (t)] = eα22 t . (8.19)

Proof. Set Z(t) = eαW (t). Then, by Ito’s forumula,

dZ(t) =12

α2eαW (t)dt +αeαW (t)dW (t). (8.20)

Problem 8.1. Show the above equation by using Ito’s formula.

Rewriting this, we have the stochastic differential equation,

dZ(t) =12

α2Z(t)dt +αZ(t)dW (t), (8.21)

Z(0) = 1. (8.22)

In integral form, we have

Z(t) = 1+12

α2∫ t

0Z(s)ds+α

∫ t

0Z(s)dW (s). (8.23)

Since E[∫ t

0 Z(s)dW (s)] = 0, taking the expectation, we have

m(t) = E[Z(t)] = 1+12

α2∫ t

0m(s)ds. (8.24)

Or equivalently,

m′(t) =12

α2m(t), (8.25)

m(0) = 1. (8.26)

By solving this, we have

E[eαW (t)] = m(t) = eα2t/2. (8.27)

Chapter 9

Differential Equations

9.1 Ordinary Differential EquationTheorem 9.1. Consider the following ordinary differential equation,

dB(t)dt

= rB(t), (9.1)

B(0) = B0. (9.2)

The solution of this equation is given by

B(t) = B0ert . (9.3)

Proof. By reordering (9.1), we have

dB(t)B(t)

= rdt. (9.4)

Integrating the both sides, we have∫ t

0

dB(s)B(s)

=∫ t

0rds. (9.5)

Thus,

logB(t) = rt +C, (9.6)

or

B(t) =Cert . (9.7)

36

9.2. GEOMETRIC BROWNIAN MOTION 37

Using the initial condition (9.2), we finally have

B(t) = B0ert . (9.8)

9.2 Geometric Brownian MotionDefinition 9.1 (Geometric Brownian Motion). A stochastic process X(t) satisfy-ing the following dynamics is called geometric brownian motion.

dX(t) = αX(t)dt +σX(t)dW (t), (9.9)X(0) = x0, (9.10)

where α is called the drift and σ is called the volatility.

Theorem 9.2 (SDE of Geometric Brownian Motion). The solution of the equation

dX(t) = αX(t)dt +σX(t)dW (t), (9.11)X(0) = x0, (9.12)

is given by

X(t) = x0e(α−σ2/2)t+σW (t). (9.13)

Proof. Here the tricks we used in Theorem 9.1 may not work here because dW (t)terms should be considered as the stochastic difference.

To simplify the proof, we assume X(t)> 0. Set

Z(t) = f (t,X(t)) = logX(t). (9.14)

Then by Ito’s formula (Theorem 7.4), we have

dZ(t) =∂ f∂ t

dt +∂ f∂x

dX(t)+12

∂ 2 f∂x2 (dX(t))2

=dX(t)X(t)

− 12(dX(t))2

X(t)2 . (9.15)

38 CHAPTER 9. DIFFERENTIAL EQUATIONS

By (9.11),

(dX(t))2 = α2X(t)2(dt)2 +2σX(t)dW (t)αX(t)dt +σ

2X(t)2(dW (t))2

= σ2X(t)2dt. (9.16)

Using this in (9.15) as well as (9.11), we have

dZ(t) = αdt +σdW (t)− 12

σ2dt. (9.17)

Luckily, the right hand side can be integrated directly here!

Z(t)−Z(0) =∫ t

0(α− 1

2σ

2)ds+σ

∫ t

0dW (s) (9.18)

= (α− 12

σ2)t +σdW (t). (9.19)

Using the initial condition (9.12), we have Z(0) = logx0. Thus,

Z(t) = (α− 12

σ2)t +σdW (t)+ logx0, (9.20)

or

X(t) = eZ(t) = x0e(α−σ2/2)t+σW (t). (9.21)

Remark 9.1. This proof has not prove the existence of the solution yet. Morerigorous treatment of the proof, see the textbook like Oksendal [2003].

9.3 Stochastic Process and Partial Differential Equa-tion

Theorem 9.3 (Feynman-Kac Represenation). Consider the following equation ofa function F = F(t,x):

∂F∂ t

+µ∂F∂x

+12

σ2 ∂ 2F

∂x2 − rF = 0, (9.22)

9.3. STOCHASTIC PROCESS AND PARTIAL DIFFERENTIAL EQUATION39

with its boundary condition,

F(T,x) = Φ(x). (9.23)

In addition, consider the following stochastic differential equation,

dX(t) = σdt +σdW (t). (9.24)

Then, the solution of (9.22) has the following Feynman-Kac representation:

F(t,x) = e−r(T−t)E[Φ(XT )|X(t) = x]. (9.25)

Chapter 10

Martingale Representation ofRandom Variables

10.1 Why the Representation Theorem?As we saw in Theorem 7.2, any stochastic integral:

X(t) =∫ t

0g(s)dW (s), (10.1)

is Ft-martingale.

Problem 10.1. So, what was martingale and why was it so?

Here we will discuss the reverse.

Problem 10.2. What is the reverse?

Problem 10.3. Why is it important?

For any random variable X , it is good to have the representation like

X−E[X ] =∫ T

0g(s)dW (s), (10.2)

for some Ft-adopted function g.In the context of mathematical finance, X is a derivative and W is the underly-

ing stock price. Having the representation like this is sometimes called complete.

Problem 10.4. What does this “complete” mean? Why is it useful in mathemati-cal finance?

40

10.2. INTUITIVE EXPLANATION 41

10.2 Intuitive ExplanationSuppose X(t) is a stochastic process adopted to a brownian motion W (t).

Problem 10.5. What was “adopted”? What will this adoptedness will affect toX(t)?

Let X(t) be a stochastic process represented by

X(t) =∫ t

0g(s)dW (s), (10.3)

or

dX(t) = g(t)dW (t). (10.4)

Recalling the approximation used in Ito-integral, we have

X(t)W (t) =∫ t

0d(X(s)W (s)) (10.5)

=∫ t

0g(s)dW (s)dW (s) (10.6)

=∫ t

0g(s)ds (10.7)

Problem 10.6. Explain the detail of this derivation using an approximation.

Thus, if we can find a function g such that

g(t) =ddt(X(t)W (t)), (10.8)

this g will satisfy the stochastic integral representation;

X(t) =∫ t

0g(s)dW (s). (10.9)

10.3 Constructing Martingale RepresentationSuppose X has a special form:

X = e∫ T

0 hdW− 12∫ T

0 h2dt , (10.10)

and set

X(t) = e∫ t

0 hdW− 12∫ t

0 h2ds, (10.11)

and check the dynamics for X .

42CHAPTER 10. MARTINGALE REPRESENTATION OF RANDOM VARIABLES

Problem 10.7. What can you say about this form? Why are we going to use thisform?

Then, by Ito’s formula 7.4, we can find the dynamic for X ,

dX = X(t)(hdW − 12

h2dt)+12

X(t)(hdW )2 = X(t)hdW (10.12)

Problem 10.8. Check the derivation of (10.12)!

Integrating (10.12) over (0, t), we have

X(t) = 1+∫ t

0X(s)hdW (10.13)

Thus, we have the martingale representation:

X = 1+∫ T

0X(t)hdW. (10.14)

In general case, we can use an approximation of the form (10.10) which isfound to be dense in L2(Ft)(see Oksendal [2003, p53]).

Chapter 11

Simple Image of Malliavin Calculus

11.1 Differential Calculus of Power SeriesProblem 11.1. What is “differential calculus”? What is “differential calculus”for?

For an example, we have a simple rule:

ddx

xn = nxn−1. (11.1)

Another simple rule:

ddx

ex = ex. (11.2)

Problem 11.2. How do you derive this?

We have a power series expansion of ex;

ex =∞

∑n=0

xn

n!(11.3)

Then we can easily derive (11.2).In general, if we have a power series expansion:

f (x) =∞

∑n=0

anxn, (11.4)

43

44 CHAPTER 11. SIMPLE IMAGE OF MALLIAVIN CALCULUS

we have

f ′(x) =∞

∑n=0

nanxn−1. (11.5)

Problem 11.3. How about∫

f (x)dx?

Problem 11.4. Can you think of other expressions of differential calculus?

11.2 Differential Calculus based on Integral by PartsProblem 11.5. What is integral by parts?

∫f (x)g′(x)dx = f (x)g(x)−

∫f ′(x)g(x)dx (11.6)

Problem 11.6. How can we use integral by parts to define differential calculus?

11.3 Differential Calculus for Random VariablesAs we saw in (11.7), we have a representation of random variable using stochasticintegral such as

X =∫ T

0g(s)dW (s). (11.7)

Problem 11.7. What is the relationship between X and the brownian motionW (t)?

Problem 11.8. What will happen to X if the trajectory of W (t) is slightly changed?

Intuitively, the Malliavin derivative of X is the derivative of X with respectto the brownian motion W (t), either by finding appropriate expansion formula(Wiener-Ito Chaos expansion) or by appropriate integral by parts formula.

Problem 11.9. How can you use Malliavin derivative?

Chapter 12

Iterated Ito-integral and ChaosExpansion

12.1 Iterated Ito-integral and Hermite polynomialsThe family of Hermite polynomials constitutes an orthogonal basis on L2 space.

hn(x) = (−1)nex2/2 dn

dxn e−x2/2. (12.1)

Problem 12.1. What is orthogonal basis? What is useful?

In particularly,

h0(x) = 1,h1(x) = x,

h2(x) = x2−1,. . .

The iterated Ito integral can be calculated by using Hermite polynomials!

n!∫ T

0

∫ tn

0. . .∫ t2

0g(t1) . . .g(tn)dW (t1) . . .dW (tn) = ||g||nhn(θ/||θ ||), (12.2)

where

θ =∫ T

0g(t)dW (t). (12.3)

Problem 12.2. What does this relation mean?

45

46 CHAPTER 12. ITERATED ITO-INTEGRAL AND CHAOS EXPANSION

12.2 Chaos ExapnasionDefine

In(g) =∫ T

0gdW (t1) . . .dW (tn). (12.4)

Let ξ be FT measurable random variable in L2. Then we have the Wiener-ItoChaos expansion such that

ξ = ∑ In( fn), (12.5)

for some unique sequence of functions fn.

Example 12.1.

ξ =W 2(T ) = T + I2(1) = T +∫ T

0

∫ t2

01dW (t1)dW (t2). (12.6)

Problem 12.3. What does this mean?

12.3 Skorohod integralSuppose u(t) has Chaos Expansion:

u(t) = ∑ In( fn,t), (12.7)

Define Skorohod integral as∫ T

0u(t)δW (t) =

∞

∑n=0

In+1( fn,t), (12.8)

where f is the symmetrization of f .

Problem 12.4. What is the difference between usual Ito-integral and Skorohodintegral?

Example 12.2. ∫ T

0W (T )δW (t) = I2(1) =W 2(T )−T. (12.9)

Problem 12.5. Any differences between∫ T

0 W (t)dW (t)?

12.4. MALLIAVIN DERIVATIVE 47

12.4 Malliavin DerivativeSuppose a random variable F has chaos expansion:

F = ∑ In( fn), (12.10)

Define the Malliavin derivative as

DtF = ∑nIn−1( fn(t)). (12.11)

12.5 Clarck-Ocone FormulaIndeed we have

F = E[F ]+∫ T

0E[DtF |Ft ]dW (t). (12.12)

Bibliography

T. Bjork. Arbitrage Theory in Continuous Time. Oxford Finance. Oxford Univ Pr,2nd edition, 2004.

G. Di Nunno, B. Oksendal, and F. Proske. Malliavin calculus for Levy processeswith Applications to Finance. Springer Verlag, 2009.

R. Durrett. Probability: Theory and Examples. Thomson Learning, 1991.

G. Folland. Real analysis: modern techniques and their applications. Wiley NewYork, 1984.

B. Oksendal. Stochastic Differential Equations: An Introduction With Applica-tions. Springer-Verlag, 2003.

H. Toyoizumi. Performance Evaluation based on Stochastic Analysis. 2009. URLhttp://www.f.waseda.jp/toyoizumi/.

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