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.