The Mathematics of Financial Modelingand Investment Management

10-StochDiffEq Page 272 Wednesday, February 4, 2004 12:51 PM

272 The Mathematics of Financial Modeling and Investment Management

t t

Zt ( ω, ) = ∫ as ( ω, ) sd + ∫ bs ( ω, ) dBs ( s ω, )

0 0

Itô processes can be written in the shorter differential form as

dZt = adt + bdBt

It should be clear that the latter formula is just a shorthand for the inte-

gral definition.

THE 1-DIMENSIONAL ITÔ FORMULA

One of the most important results concerning Itô processes is a formula

established by Itô that allows one to explicitly write down an Itô process

which is a function of another Itô process. Itô’s formula is the stochastic

equivalent of the change-of-variables formula of ordinary integration.

We will proceed in two steps. First we will introduce Itô’s formula for

functions of Brownian motion and then for functions of general Itô pro-

cesses. Suppose that the function g(t,x) is twice continuously differentia-

ble in [0,∞ ) × R and that Bt is a one-dimensional Brownian motion. The

function Yt = g(t,Bt) is a stochastic process. It can be demonstrated that

the process Yt = g(t,Bt) is an Itô process of the following form

∂ g 1 ∂^2 g  ∂ g

dYt =  ------( tBt) + ------------( tBt) dt + ------( tBt) dBt

∂ t

,

(^2) ∂ x^2
,
 ∂ x
,
The above is Itô’s formula in the case the underlying process is a Brown-
ian motion. For example, let’s suppose that g(t,x) = x^2. In this case we
can write
∂ g ∂ g ∂
2
g
------= 0 , ------ = 2 x , --------- = 2
∂ t ∂ x ∂ x^2
2
Inserting the above in Itô’s formula we see that the process Bt can be
represented as the following Itô process
dYt = dt + 2BtdBt
or, explicitly in integral form