Hello, Quant masters. Although my background is in finance, I am not well versed in quant finance. Right now I am taking a quant finance course and have a problem with the following problem:Let Wt be a Brownian motion and let mk(t) = E(Wt^k), k=0,1,2....a) Use Ito's formula to show that mk(t) = [k(k-1)]/2 times integral from 0 to t of Mk-2(s)ds for k>=2b) Show that E(Wt^(2k)) = [(2k)!t^k ]/(2^k)k!c) Show that E(Wt^(2k+1))= 0I know that I have to use Ito's formula with the function g(x)=x^k and the fact that a stochastic integral is a martingale and hence has zero expectation. However, I can't get past that. Please help a fellow quant newbie with some knowledge!

For any positive interger N, letm(N,t) = E( (W_t)^N )The use Ito's formula, you can getdm(N,t) = N(N-1)/2 * m(N-2,t)dtWhen N=1, m(1,t) = 0. So for all odd N, m(N,t) = 0. For N=0, E(0,t) =1. So mathematical induction provids the result as you have stated for even N.