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
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