CHAPTER 1

INTRODUCTION

The notion of multiple stochastic integration was first introduced

by Norbert Wiener in connection with his studies of the Brownian Motion

process. Given any 1-dimensional Brownian path X(t) and any function

f(t) which depends only on t, he wanted to be able to define the integral

T

f(t)dX(t) = 1(f); however, the integral makes no sense as a Stieltjes sum

0

since X(t) is not a function of bounded variation.

The problem becomes even more complex when one wants to consider

multiple stochastic integrals

••• f f(t.,...,t ) dXCO-.-dXCt ) = I (f)

J _i.n I 1 n 1 n n

J It J

where f is a real valued function defined on

]R+U.

K. Ito [3] was the

first to establish the existence of I (f) for symmetric functions f onIR+*

2

whose L -norms are finite. Using these multiple integrals, K. Ito was able

2

to establish an orthogonal decomposition of the space of L -Brownian

functionals. This decomposition agrees exactly with one obtained by R.

Cameron and W. Martin [1] using Hermite polynomials and clarifies much of

the work started by N. Wiener in "The Homogeneous Chaos" [9].

To outline these results let (ft,®,Pr) be a Lebesgue probability

space and (R,f,m) the real numbers with Lebesgue measure m. A Brownian

Received by the editors June 1980. This paper is a revised version

of the author's doctoral dissertation at Yale University, 1979, which

was supported by a Yale University graduate fellowship.

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