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. 1

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