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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