MULTIPLE STOCHASTIC INTEGRAL

(1) E(X(I)) = 0 for all I e ? , and

(2) X(I)H

X(J)

whenever I 0 J = fr.

Keeping these properties in mind we now define the Multiple Wiener

Integral. Let S( Q ) be the set of all real valued functions of

n-variables f(t,,...,t ) for which f(t_,...,t ) = 0 unless 0 t., •••

I n I n — 1

•••t _ _ T. A step function in this class is a function g defined by

g(tr...,tn)=c iff (tl....,t) el » " * h

I n I n

for some choice of disjoint intervals I.,...,I with I. I... and some

J 1' p l l+l

choice of c. . where li , ! « • • • ! p. Then we can define

l •• • l — 1 2 n —

1 n

- . { 8(t1,...,tn)dX(t1)...dX(t ) = 2 .

Ci...iX(Ii-

J

1± •• • i p 1 n 1

0tn«««t T 1 n-

1 n~

•••X(I± ).

n

Now since I.,...,I are disjoint, properties (1) and (2) imply

1 P

I 12

Ci

...i

X(Ii )"'X(Ii H22

l£i1---in£P 1 n 1 n L (ft)

2-rf Ci ...i m(Ii "#'in(Iin

li •••! p Al n 1

)

|8||22

n

L

Z

(lR

n

)

We can therefore extend the definition of the multiple Wiener integral to

all g e

S(Qn)O L2(lRn)

as we did before: