10-StochDiffEq Page 267 Wednesday, February 4, 2004 12:51 PM

CHAPTER

10

Stochastic Differential Equations

C

hapter 8 introduced stochastic integrals, a mathematical concept

used for defining stochastic differential equations, the subject of this

chapter. Stochastic differential equations solve the problem of giving

meaning to a differential equation where one or more of its terms are

subject to random fluctuations. For instance, consider the following

deterministic equation:

dy

------ = ft()y

dt

We know from our discussion on differential equations (Chapter 9)

that, by separating variables, the general solution of this equation can

be written as follows:

y = A exp[ ∫ ft()td ]

A stochastic version of this equation might be obtained, for instance, by

perturbing the term f, thus resulting in the “stochastic differential equa-

tion”

------ = [ft+

dy

() ε]dt

y

where εis a random noise process.

As with stochastic integrals, in defining stochastic differential equa-

tions it is necessary to adopt an ensemble view: The solution of a stochas-

tic differential equation is a stochastic process, not a single function. We

267