MULTIPLE STOCHASTIC INTEGRAL 7
COROLLARY 1.3 If g e Q then g is exactly of the form
nn+-**+np=n p ^ 0t,*--t
1 I n .
t, el. lknI.
k
I
Proofs of the above mentioned results can be found in K. Ito [3],
S. Kakutani [6],and R. Cameron and W. Martin [1]. The purpose of this
paper is to describe a unified approach to understanding these and other
theorems for a broader class of stochastic processes than the Brownian
Motion alone. The simplest alternate process is the Poisson process
which is used in Chapter 3 as motivation for the main theorem (Chapter
4). Chapter 3 is essentially the theory of Discrete Chaos introduced
by N. Wiener [9] and N. Wiener and A. Wintner [10].
This paper views the multiple stochastic integral as a countably
2
additive extension of the finitely additive L fo)-valued set function
X defined for elementary subsets of S. X is, in fact, well
defined for all elementary subsets of the n-cube [0,T] . The proof of
the fact that the set function X has a countably additive extension
to the o-field of all Borel subsets of [0,T] is in no way straight-
forward. The proof of the existence of such an extension consumes most
of Chapter 4 and represents the main theorem of this paper. Once this
result is obtained and combined with some combinatorial lemmas given
in Chapter 5, the connection between multiple stochastic integration,
Hermite polynomials, and the homogeneous chaos is obvious.
In the time since I originally wrote this manuscript, I have become
aware of the works of many other authors which address the same problems
that I have. Most of the results of this paper can be derived by using
the techniques developed by Paul Meyer [8]. He takes a different view-
point of the problem by exploiting the martingale properties of the
Previous Page Next Page