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Stochastic Processes

Financial variables, such as prices and returns, are random time-

dependent variables. The notion of stochastic process is used to de-

scribe their behavior. Specifically, the Wiener process (or the Brownian

motion) plays the central role in mathematical finance. Section 4.1

begins with the generic path: Markov process ! Chapmen-Kolmo-

gorov equation ! Fokker-Planck equation ! Wiener process. This

methodology is supplemented with two other approaches in Section

4.2. Namely, the Brownian motion is derived using the Langevinâ€™s

equation and the discrete random walk. Then the basics of stochastic

calculus are described. In particular, the stochastic differential equa-

tion is defined using the Itoâ€™s lemma (Section 4.3), and the stochastic

integral is given in both the Ito and the Stratonovich forms

(Section 4.4). Finally, the notion of martingale, which is widely popu-

lar in mathematical finance, is introduced in Section 4.5.

4.1 MARKOV PROCESSES

Consider a process X(t) for which the values x1 , x2 , . . . are meas-

ured at times t1 , t2 , . . . Here, one-dimensional variable x is used

for notational simplicity, though extension to multidimensional

systems is trivial. It is assumed that the joint probability density

f(x1 , t1 ; x2 , t2 ; . . . ) exists and defines the system completely. The con-

ditional probability density function is defined as

29

30 Stochastic Processes

f(x1 , t1 ; x2 , t2 ; . . . xk , tk jxkÃ¾1 , tkÃ¾1 ; xkÃ¾2 , tkÃ¾2 ; . . . ) Â¼

f(x1 , t1 ; x2 , t2 ; . . . xkÃ¾1 , tkÃ¾1 ; . . . )=f(xkÃ¾1 , tkÃ¾1 ; xkÃ¾2 , tkÃ¾2 ; . . . ) (4:1:1)

In (4.1.1) and further in this section, t1 > t2 > . . . tk > tkÃ¾1 > . . .

unless stated otherwise. In the simplest stochastic process, the present

has no dependence on the past. The probability density function for

such a process equals

Y

f(x1 , t1 ; x2 , t2 ; . . . ) Â¼ f(x1 , t1 )f(x2 , t2 ) . . . f(xi , ti ) (4:1:2)

i

The Markov process represents the next level of complexity, which

embraces an extremely wide class of phenomena. In this process, the

future depends on the present but not on the past. Hence, its condi-

tional probability density function equals

f(x1 , t1 ; x2 , t2 ; . . . xk , tk jxkÃ¾1 , tkÃ¾1 ; xkÃ¾2 , tkÃ¾2 ; . . . ) Â¼

f(x1 , t1 ; x2 , t2 ; . . . xk , tk jxkÃ¾1 , tkÃ¾1 ) (4:1:3)

This means that evolution of the system is determined with the initial

condition (i.e., with the value xkÃ¾1 at time tkÃ¾1 ). It follows for the

Markov process that

f(x1 , t1 ; x2 , t2 ; x3 , t3 ) Â¼ f(x1 , t1 jx2 , t2 )f(x2 , t2 jx3 , t3 ) (4:1:4)

Using the definition of the conditional probability density, one can

introduce the general equation

Ã°

f(x1 , t1 jx3 , t3 ) Â¼ f(x1 , t1 ; x2 , t2 jx3 , t3 )dx2

Ã°

Â¼ f(x1 , t1 jx2 , t2 ; x3 , t3 )f(x2 , t2 jx3 , t3 )dx2 (4:1:5)

For the Markov process,

f(x1 , t1 jx2 , t2 ; x3 , t3 ) Â¼ f(x1 , t1 jx2 , t2 ), (4:1:6)

Then the substitution of equation (4.1.6) into equation (4.1.5) leads to

the Chapmen-Kolmogorov equation

Ã°

f(x1 , t1 jx3 , t3 ) Â¼ f(x1 , t1 jx2 , t2 )f(x2 , t2 jx3 , t3 )dx2 (4:1:7)

This equation can be used as the starting point for deriving the

Fokker-Planck equation (see, e.g., [1] for details). First, equation

(4.1.7) is transformed into the differential equation

31

Stochastic Processes

1 @2

@ @

f(x, tjx0 , t0 ) Â¼Ã€ [A(x, t)f(x, tjx0 , t0 )] Ã¾ [D(x, t)f(x, tjx0 , t0 )]Ã¾

2 @x2

@t @x

Ã°

[R(xjz, t)f(z, tjx0 , t0 ) Ã€R(zjx, t)f(x, tjx0 , t0 )]dz (4:1:8)

In (4.1.8), the drift coefficient A(x, t) and the diffusion coefficient

D(x, t) are equal

Ã°

1

A(x, t) Â¼ lim (z Ã€ x)f(z, t Ã¾ Dtjx, t)dz (4:1:9)

Dt!0 Dt

Ã°

1

(z Ã€ x)2 f(z, t Ã¾ Dtjx, t)dz

D(x, t) Â¼ lim (4:1:10)

Dt!0 Dt

The integral in the right-hand side of the Chapmen-Kolmogorov

equation (4.1.8) is determined with the function

1

R(xjz, t) Â¼ lim f(x, t Ã¾ Dtjz, t) (4:1:11)

Dt!0 Dt

It describes possible discontinuous jumps of the random variable. Neg-

lecting this term in equation (4.1.8) yields the Fokker-Planck equation

@ @

f(x, tjx0 , t0 ) Â¼ Ã€ [A(x, t)f(x, tjx0 , t0 )]

@t @x

(4:1:12)

1 @2

Ã¾ [D(x, t)f(x, tjx0 , t0 )]

2 @x2

This equation with A(x, t) Â¼ 0 and D Â¼ const is reduced to the

diffusion equation that describes the Brownian motion

D @2

@

f(x, tjx0 , t0 ) Â¼ f(x, tjx0 , t0 ) (4:1:13)

2 @x2

@t

Equation (4.1.13) has the analytic solution in the Gaussian form

f(x, tjx0 , t0 ) Â¼ [2pD(t Ã€ t0 )]Ã€1=2 exp [Ã€(x Ã€ x0 )2 =2D(t Ã€ t0 )] (4:1:14)

Mean and variance for the distribution (4.1.14) equal

E[x(t)] Â¼ x0 , Var[x(t)] Â¼ E[(x(t) Ã€ x0 )2 ] Â¼ s2 Â¼ D(t Ã€ t0 ) (4:1:15)

The diffusion equation (4.1.13) with D Â¼ 1 describes the standard

Wiener process for which

E[(x(t) Ã€ x0 )2 ] Â¼ t Ã€ t0 (4:1:16)

32 Stochastic Processes

The notions of the generic Wiener process and the Brownian motion

are sometimes used interchangeably, though there are some fine

differences in their definitions [2, 3]. I shall denote the Wiener process

with W(t) and reserve this term for the standard version (4.1.16), as it

is often done in the literature.

The Brownian motion is the classical topic of statistical physics.

Different approaches for introducing this process are described in the

next section.

4.2 BROWNIAN MOTION

In mathematical statistics, the notion of the Brownian motion is

used for describing the generic stochastic process. Yet, this term

referred originally to Brownâ€™s observation of random motion of

pollen in water. Random particle motion in fluid can be described

using different theoretical approaches. Einsteinâ€™s original theory of

the Brownian motion implicitly employs both the Chapman-Kolmo-

gorov equation and the Fokker-Planck equation [1]. However, choos-

ing either one of these theories as the starting point can lead to the

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