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

Dt!0

[DW(t)=Dt)] ¼ lim

Dt!0

[Dt

1=2

](4:2:16)

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

form

3

dy(t) ¼ mdt þ sdW(t) (4:3:1)

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

Note that the term dW(t) ¼ [W(t þ dt) W(t)] has the following

properties

E[dW] ¼ 0, E[dW dW] ¼ dt, E[dW dt] ¼ 0(4:3:2)

Let us calculate (dy)

2

having in mind (4.3.2) and retaining the terms

O(dt):

4

(dy)

2

¼ [mdt þ sdW]

2

¼ m

2

dt

2

þ 2mdt sdW þ s

2

dW

2

s

2

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 Ito’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)

dF(y, t) ¼

@F

@y

dy þ

@F

@t

dt þ

1

2

@

2

F

@y

2

(dy)

2

¼

@F

@y

dy þ

@F

@t

þ

s

2

2

@

2

F

@ y

2

dt (4:3:4)

The Ito’s expression (4.3.4) has an additional term in comparison with

the differential for a function with deterministic independent vari-

Stochastic Processes 35

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