The Mathematics of Financial Modelingand Investment Management

10-StochDiffEq Page 269 Wednesday, February 4, 2004 12:51 PM

Stochastic Differential Equations 269

The continuous-time limit of the random walk is the Brownian

motion. However the paths of a Brownian motion are not differentiable.

As a consequence, it is not possible to take the continuous-time limit of

first differences and to define the white noise process as the derivative of

a Brownian motion. In the domain of ordinary functions in continuous

time, the white noise process can be defined only through its integral,

which is the Brownian motion. The definition of stochastic differential

equations must therefore be recast in integral form.

A sensible definition of a stochastic differential equation must

respect a number of constraints. In particular, the solution of a stochas-

tic differential equation should be a “perturbation” of the associated

deterministic equation. In the above example, for instance, we want the

solution of the stochastic equation

------ = [ft+

dy

() ε(tω , )]dt

dy

to be a perturbation of the solution

y = A exp( ∫ ft()td )

of the associated deterministic equation

dy

------ = ft()dt

y

In other words, the solution of a stochastic differential equation should

tend to the solution of the associated deterministic equation in the limit

of zero noise. In addition, the solutions of a stochastic differential equa-

tion should be the continuous-time limit of some discrete-time process

obtained by discretization of the stochastic equation.

A formal solution of this problem was proposed by Kyosi Itô in the

1940s and, in a different setting, by Ruslan Stratonovich in the 1960s.

Itô and Stratonovich proposed to give meaning to a stochastic differen-

tial equation through its integral equivalent. The Itô definition proceeds

in two steps: in the first step, Itô processes are defined; in the second

step, stochastic differential equations are defined.

■ Step 1: Definition of Itô processes. Given two functions φ(tω , ) and

ψ(tω , ) that satisfy usual conditions to be defined later, an Itô pro-

cess—also called a stochastic integral—is a stochastic process of the

form: