MULTIPLE STOCHASTIC INTEGRAL
= £ a X(I
A - X d )"k
n1+-**+n1 n I k K
k = l , 2 , . . . , n
n. e 3N
IP i s called the Polynomial Chaos of degree n.
Note ( i i i ) implies that we may assume I , . . . , 1 are disjoin t in the presen-
tatio n of g. I t i s clear tha t IP i s a linea r subspace of L (ft) and that
^ for n = 0 , 1 , 2 , . . . . If we denote b y ^ ? t h e ring of a l l such
polynomials we have &t = U W . The closure of £ft in L (ft), denoted
is the space of L -Brownian Functionals.
Let us put Q = f the constant functions in L (ft). Furthermore,
let Q = T © I P . , the orthocomplement ofP . , inI P . This means that
n n n-1 n-1 n
g M iff g £ P and g J f for all f ep _ . Q is called the nC
n n -- 1 n-1 n
Homogeneous Chaos. By definition^??= ® Q .
2 n=0 n
Let H (u,t) = -—f- e —IT (e ^t) denote the n generalized
n n. du
Hermite polynomial. The linear subspaces Q have a particularly nice
representation in terms of these polynomials.
THEOREM 1.1 Any g e Q can be represented in the form
8(10)= S a
...n "frn (X(I .ui), m(I ))
n«+## •+n =1 1 1 p i=l i
where I-,...,I are disjoint elements of J- .
Proof: See S. Kakutani .
The Multiple Wiener Integral
We now consider the multiple Wiener integral as defined by K. Ito 
and see how this applies to the homogeneous chaos of the preceding section.
To begin with / f(t)dX(t) can be defined easily when f is the characteristic
function of a set I C [0,T] , I ej by