MULTIPLE STOCHASTIC INTEGRAL

3

g(u))

= £ a X(I

A - X d )"k

.

n1+-**+n1 n I k K

1 kr-

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-

2

tatio n of g. I t i s clear tha t IP i s a linea r subspace of L (ft) and that

^

n

^- ^

n +

^ for n = 0 , 1 , 2 , . . . . If we denote b y ^ ? t h e ring of a l l such

00

polynomials we have &t = U W . The closure of £ft in L (ft), denoted

n=U n

is the space of L -Brownian Functionals.

2

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

u_ 2

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

...n "frn (X(I .ui), m(I ))

n«+## •+n =1 1 1 p i=l i

1 p

F

where I-,...,I are disjoint elements of J- .

1 P

Proof: See S. Kakutani [6].

The Multiple Wiener Integral

We now consider the multiple Wiener integral as defined by K. Ito [3]

and see how this applies to the homogeneous chaos of the preceding section.

T

To begin with / f(t)dX(t) can be defined easily when f is the characteristic

0

function of a set I C [0,T] , I ej by