8-Stochastic Integrals Page 217 Wednesday, February 4, 2004 12:50 PM

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CHAPTER

8

Stochastic Integrals

n Chapter 4, we explained definite and indefinite integrals for deter-

ministic functions. Recall that integration is an operation performed

on single, deterministic functions; the end product is another single,

deterministic function. Integration defines a process of cumulation: The

integral of a function represents the area below the function. However,

the usefulness of deterministic functions in economics and finance the-

ory is limited. Given the amount of uncertainty, few laws in economics

and finance theory can be expressed through them. It is necessary to

adopt an ensemble view, where the path of economic variables must be

considered a realization of a stochastic process, not a deterministic

path. We must therefore move from deterministic integration to stochas-

tic integration. In doing so we have to define how to cumulate random

shocks in a continuous-time environment. These concepts require rigor-

ous definition. This chapter defines the concept and the properties of

stochastic integration. Based on the concept of stochastic integration,

Chapter 10 defines stochastic differential equations.

Two observations are in order:

■ While ordinary integrals and derivatives operate on functions and

yield either individual numbers or other functions, stochastic integra-

tion operates on stochastic processes and yield either random vari-

ables or other stochastic processes. Therefore, while a definite

integral is a number and an indefinite integral is a function, a stochas-

tic integral is a random variable or a stochastic process. A differential

equation—when equipped with suitable initial or boundary condi-

tions—admits as a solution a single function while a stochastic differ-

ential equations admits as a solution a stochastic process.

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