The Mathematics of Financial Modelingand Investment Management

10-StochDiffEq Page 271 Wednesday, February 4, 2004 12:51 PM

Stochastic Differential Equations 271

Note that in this approximation the functions φ *( tX, ) , ψ *( tX, ) will

not coincide with the functions φ( tX, ) , ψ( tX, ). Using the latter would

(in general) result in a poor approximation.

The following sections will define Itô processes and stochastic dif-

ferential equations and study their properties.

ITÔ PROCESSES

Let’s now formally define Itô processes and establish key properties, in

particular the Itô formula. In the previous section we stated that an Itô

process is a stochastic process of the form

t t

Zt( ω , ) = ∫ as( ω , ) ds+ ∫ bs( ω , ) dsB( ω , )

0 0

To make this definition rigorous, we have to state the conditions

under which (1) the integrals exist and (2) there is no anticipation of

information. Note that the two functions aand bmight represent two

stochastic processes and that the Riemann-Stieltjes integral might not

exist for the paths of a stochastic process. We have therefore to demon-

strate that both the Itô integral and the ordinary integral exist. To this

end, we define Itô processes as follows.

Suppose that a 1-dimensional Brownian motion Bt is defined on a

probability space (Ω ,ℑ ,P) equipped with a filtration ℑ t. The filtration

might be given or might be generated by the Brownian motion Bt. Sup-

pose that both aand bare adapted to ℑ tand jointly measurable in ℑ × R.

Suppose, in addition, that the following two integrability conditions hold:

t

Pb∫^2 ( sω , ) ds< ∞ for all t≥ 0 = 1

0

and

t

P ∫ as( ω , )ds< ∞ for all t≥ 0 = 1

0

These conditions ensure that both integrals in the definition of Itô pro-

cesses exist and that there is no anticipation of information. We can

therefore define the Itô process as the following stochastic process: