4.8. Let In(t,) = tn/2Hn(W(t,)/ t), where Hn is the nth Hermite polynomial, (see (Kreyszig 1999,...

4.8. Let In(t,ω) = tn/2Hn(W(t,ω)/

√

t), where Hn is the nth Hermite polynomial,

(see (Kreyszig 1999, pp. 246–247) or (Abramowitz and Stegun 1965, Chap. 22)), and the first few Hermite polynomials are H0(x) = 1, H1(x) = x, H2(x) = x2−1, and H3(x) = x3−3x. What are I0(t,ω),. . . ,I3(t,ω)? The aim of this exercise is to show that under Ito integration the functions In(t) are the stochastic analogue of powers under ordinary integration. Use Ito’s formula to show that T

0 In−1(t,ω)dW(t,ω) =

1 n

In(T,ω) .

See how Ito integration maps In−1 to 1n In just as ordinary integration analogously maps the power tn−1 to 1n tn. Continue to hence deduce

· · ·

0≤t1≤···≤tn≤t 1dW(t1,ω)dW(t2,ω) . . . dW(tn,ω)

=

1 n!

tn/2Hn(W(t,ω)/

√

t) .

Note: Hn(x), among other properties, satisfy the recurrences H



n = nHn−1 and H



n−xH



n+nHn = 0 .