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
f(t)dX(t) = 1(f); however, the integral makes no sense as a Stieltjes sum
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
K. Ito [3] was the
first to establish the existence of I (f) for symmetric functions f onIR+*
whose L -norms are finite. Using these multiple integrals, K. Ito was able
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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