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
•• 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:
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