The Mathematics of Financial Modelingand Investment Management

8-Stochastic Integrals Page 220 Wednesday, February 4, 2004 12:50 PM

220 The Mathematics of Financial Modeling and Investment Management

b

W = ∫XtdBt

a

or

b

W = ∫XtºdBt

a

is a random variable W associated to a stochastic process if the time

interval is fixed or, if the time interval is variable, is another stochastic

process Wt. The stochastic integral represents the cumulation of the sto-

chastic products XtdBt. As we will see in Chapter 10, the rationale for

this approach is that we need to represent how random shocks feed back

into the evolution of a process. We can cumulate separately the deter-

ministic increments and the random shocks only for linear processes. In

nonlinear cases, as in the simple example of the bank account, random

shocks feed back into the process. For this reason we define stochastic

integrals as the cumulation of the product of a process X by the random

increments of a Brownian motion.

Consider a stochastic process Xt over an interval [S,T]. Recall that a

stochastic process is a real variable X(ω)t that depends on both time and

the state of the economy ω. For any given ω, X(⋅)t is a path of the process

from the origin S to time T. A stochastic process can be identified with

the set of its paths equipped with an appropriate probability measure. A

stochastic integral is an integral associated to each path; it is a random

variable that associates a real number, obtained as a limit of a sum, to

each path. If we fix the origin and let the interval vary, then the stochas-

tic integral is another stochastic process.

It would seem reasonable, prima facie, to define the stochastic inte-

gral of a process X(ω)t as the definite integral in the sense of Rieman-

Stieltjes associated to each path X(⋅)t of the process. If the process X(ω)t

has continuous paths X(⋅,ω), the integrals

T

W () ω = ∫Xs( ω , ) sd

S

exist for each path. However, as discussed in the previous section, this is

not the quantity we want to represent. In fact, we want to represent the

cumulation of the stochastic products XtdBt. Defining the integral