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riving the Fokker-Planck equation. He considered one-dimensional

motion of a spherical particle of mass m and radius R that is subjected

to two forces. The first force is the viscous drag force described by the

dr

Stokes formula, F Â¼ Ã€6pZRv, where Z is viscosity and v Â¼ is the

dt

particle velocity. Another force, Z, describes collisions of the water

molecules with the particle and therefore has a random nature. The

Langevin equation of the particle motion is

dv

Â¼ Ã€6pZRv Ã¾ Z

m (4:2:1)

dt

Let us multiply both sides of equation (4.2.1) by r. Since

dv d 1d 2

r Â¼ (rv) Ã€ v2 and rv Â¼ (r ), then

dt dt 2dt

dr 2

1 d2 2 d

Â¼ Ã€3pZR (r2 ) Ã¾ Zr

m 2 (r ) Ã€ m (4:2:2)

2 dt dt dt

Note that the mean kinetic energy of a spherical particle, E[ 1 mv2 ],

2

3

equals 2 kT. Since E[Zr] Â¼ 0 due to the random nature of Z, averaging

of equation (4.2.2) yields

33

Stochastic Processes

d2 d

m 2 E[r2 ] Ã¾ 6pZR E[r2 ] Â¼ 6kT (4:2:3)

dt dt

The solution to equation (4.2.3) is

d

E[r2 ] Â¼ kT=(pZR) Ã¾ C exp (Ã€6pZRt=m) (4:2:4)

dt

where C is an integration constant. The second term in equation

(4.2.4) decays exponentially and can be neglected in the asymptotic

solution. Then

E[r2 ] Ã€ r2 Â¼ [kT=(pZR)]t (4:2:5)

0

where r0 is the particle position at t Â¼ 0. It follows from the compari-

son of equations (4.2.5) and (4.1.15) that D Â¼ kT=(pZR).1

The Brownian motion can be also derived as the continuous limit

for the discrete random walk (see, e.g., [3]). First, let us introduce the

process e(t) that is named the white noise and satisfies the following

conditions

E[e(t)] Â¼ 0; E[e2 (t)] Â¼ s2 ; E[e(t) e(s)] Â¼ 0, if t 6Â¼ s: (4:2:6)

Hence, the white noise has zero mean and constant variance s2 . The

last condition in (4.2.6) implies that there is no linear correlation

between different observations of the white noise. Such a model repre-

sents an independently and identically distributed process (IID) and is

sometimes denoted IID(0, s2 ). The IID process can still have non-

linear correlations (see Section 5.3). The normal distribution N(0, s2 )

is the special case of the white noise. First, consider a simple discrete

process

y(k) Â¼ y(k Ã€ 1) Ã¾ e(k) (4:2:7)

where the white noise innovations can take only two values2

D, with probability p, p Â¼ const < 1

e(k) Â¼ (4:2:8)

Ã€D, with probability (1 Ã€ p)

Now, let us introduce the continuous process yn (t) within the time

interval t 2 [0, T], such that

yn (t) Â¼ y([t=h]) Â¼ y([nt=T]), t 2 [0, T] (4:2:9)

34 Stochastic Processes

In (4.2.9), [x] denotes the greatest integer that does not exceed x. The

process yn (t) has the stepwise form: it is constant except the moments

t Â¼ kh, k Â¼ 1, . . . , n. Mean and variance of the process yn (T) equal

E[yn (T)] Â¼ n(2p Ã€ 1)D Â¼ T(2p Ã€ 1)D=h (4:2:10)

Var[yn (T)] Â¼ nD2 Â¼ TD2 =h (4:2:11)

Both mean (4.2.10) and variance (4.2.11) become infinite in the

limiting case h ! 0 with arbitrary D. Hence, we must impose a rela-

tion between D and h that ensures the finite values of the moments

E[yn (T)] and Var[yn (T)]. Namely, let us set

pï¬ƒï¬ƒï¬ƒ pï¬ƒï¬ƒï¬ƒ

p Â¼ (1 Ã¾ m h=s)=2, D Â¼ s h (4:2:12)

where m and s are some parameters. Then

E[yn (T)] Â¼ mT, Var[yn (T)] Â¼ s2 T (4:2:13)

It can be shown that yn (T) converges to the normal distribution

N(mT, s2 T) in the continuous limit. Hence, m and s are the drift

and diffusion parameters, respectively. Obviously, the drift parameter

differs from zero only when p 6Â¼ 0:5, that is when there is preference

for one direction of innovations over another. The continuous process

defined with the relations (4.2.13) is named the arithmetic Brownian

motion. It is reduced to the Wiener process when m Â¼ 0 and s Â¼ 1.

Note that in a more generic approach, the time intervals between

observations of y(t) themselves represent a random variable [4, 5].

While this process (so-called continuous-time random walk) better

resembles the market price variations, its description is beyond the

scope of this book.

In the general case, the arithmetic Brownian motion can be ex-

pressed in the following form

y(t) Â¼ m(t)t Ã¾ s(y(t), t)W(t) (4:2:14)

The random variable in this process may have negative values. This

creates a problem for describing prices that are essentially positive.

Therefore, the geometric Brownian motion Y(t) Â¼ exp [y(t)] is often

used in financial applications.

One can simulate the Wiener process with the following equation

pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ

[W(t Ã¾ Dt) Ã€ W(t)] DW Â¼ N(0, 1) Dt (4:2:15)

35

Stochastic Processes

While the Wiener process is a continuous process, its innovations are

random. Therefore, the limit of the expression DW=Dt does not

converge when Dt ! 0. Indeed, it follows for the Wiener process that

lim [DW(t)=Dt)] Â¼ lim [DtÃ€1=2 ] (4:2:16)

Dt!0 Dt!0

As a result, the derivative dW(t)/dt does not exist in the ordinary

sense. Thus, one needs a special calculus to describe the stochastic

processes.

4.3 STOCHASTIC DIFFERENTIAL EQUATION

The Brownian motion (4.2.14) can be presented in the differential

form3

dy(t) Â¼ mdt Ã¾ sdW(t) (4:3:1)

The equation (4.3.1) is named the stochastic differential equation.

Note that the term dW(t) Â¼ [W(t Ã¾ dt) Ã€ W(t)] has the following

properties

E[dW] Â¼ 0, E[dW dW] Â¼ dt, E[dW dt] Â¼ 0 (4:3:2)

Let us calculate (dy)2 having in mind (4.3.2) and retaining the terms

O(dt):4

(dy)2 Â¼ [mdt Ã¾ sdW]2 Â¼ m2 dt2 Ã¾ 2mdt sdW Ã¾ s2 dW2 % s2 dt (4:3:3)

It follows from (4.3.3) that while dy is a random variable, (dy)2 is a

deterministic one. This result allows one to derive the Itoâ€™s lemma.

Consider a function F(y, t) that depends on both deterministic, t, and

stochastic, y(t), variables. Let us expand the differential for F(y, t)

into the Taylor series retaining linear terms and bearing in mind

equation (4.3.3)

1 @2F

@F @F

(dy)2

dF(y, t) Â¼ dy Ã¾ dt Ã¾

2 @y2

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