The Mathematics of Financial Modelingand Investment Management

8-Stochastic Integrals Page 237 Wednesday, February 4, 2004 12:50 PM

Stochastic Integrals 237

It() ω =

t

∫ fBd t

0

It can be demonstrated that a continuous version of this process exists.

The following three properties can be demonstrated from the definition

of integral:

t

∫ dBs = Bt

0

t

∫ t –^

t

sBd s = tB ∫ B dss

0 0

t

∫

1 2 1

B dBs s = ---Bt – ---t

2 2

0

The last two properties show that, after performing stochastic integra-

tion, deterministic terms might appear.

SUMMARY

■ Stochastic integration provides a coherent way to represent that instan-

taneous uncertainty (or volatility) cumulates over time. It is thus funda-

mental to the representation of financial processes such as interest

rates, security prices or cash flows as well as aggregate quantities such

as economic output.

■ Stochastic integration operates on stochastic processes and produces

random variables or other stochastic processes.

■ Stochastic integration is a process defined on each path as the limit of a

sum. However, these sums are different from the sums of the Riemann-

Lebesgue integrals because the paths of stochastic processes are gener-

ally not of bounded variation.

■ Stochastic integrals in the sense of Itô are defined through a process of

approximation.

■ Step 1 consists in defining Brownian motion, which is the continuous

limit of a random walk.