The Mathematics of Financial Modelingand Investment Management

8-Stochastic Integrals Page 221 Wednesday, February 4, 2004 12:50 PM

Stochastic Integrals 221

b

W = ∫XtdBt

a

pathwise in the sense of Rieman-Stieltjes would be meaningless because

the paths of a Brownian motion are not of finite variation. If we define

stochastic integrals simply as the limit of XtdBt sums, the stochastic

integral would be infinite (and therefore useless) for most processes.

However, Brownian motions have bounded quadratic variation.

Using this property, we can define stochastic integrals pathwise through

an approximation procedure. The approximation procedure to arrive at

such a definition is far more complicated than the definition of the Rie-

man-Stieltjes integrals. Two similar but not equivalent definitions of sto-

chastic integral have been proposed, the first by the Japanese

mathematician Kyosi Itô in the 1940s, the second by the Russian physi-

cist Ruslan Stratonovich in the 1960s. The definition of stochastic inte-

gral in the sense of Itô or of Stratonovich replaces the increments ∆xi

with the increments ∆Bi of a fundamental stochastic process called

Brownian motion. The increments ∆Bi represent the “noise” of the pro-

cess.^1 The definition proceeds in the following three steps:

■ Step 1. The first step consists in defining a fundamental stochastic pro-

cess—the Brownian motion. In intuitive terms, a Brownian motion

Bt(ω) is a continuous limit (in a sense that will be made precise in the

following sections) of a simple random walk. A simple random walk is

a discrete-time stochastic process defined as follows. A point can move

one step to the right or to the left. Movement takes place only at dis-

crete instants of time, say at time 1,2,3,.... At each discrete instant, the

point moves to the right or to the left with probability ¹
₂.

The random walk represents the cumulation of completely uncer-

tain random shocks. At each point in time, the movement of the point

is completely independent from its past movements. Hence, the

Brownian motion represents the cumulation of random shocks in the

limit of continuous time and of continuous states. It can be demon-

strated that a.s. each path of the Brownian motion is not of bounded

total variation but it has bounded quadratic variation.

(^1) The definition of stochastic integrals can be generalized by taking a generic square
integrable martingale instead of a Brownian motion. Itô defined stochastic integrals
with respect to a Brownian motion. In 1967 H. Kunita and S. Watanabe extended
the definition of stochastic integrals to square integrable martingales.