Let W be a standard Brownian motion. Use Itô’s formula and induction over $n\in \mathbb{N}$ to calculate $E[W^n_t]$

Question

Let $W$ be a standard Brownian motion. Use Itô’s formula and induction over $n\in\mathbb{N}$ to calculate $E[W^n_t]$.

I know Itô's lemma is $f(X_t)=f(X_0)+\int_0^tf'(X_s)dX_s +0.5\int_0^tf''(X_s)d\langle X\rangle_s$ but I have no clue how to calculate what is asked.

Answer 1

Since $W_t$ is a centred and normally distributed, all its odd moments are zero. Set $n = 2k$, and use Itô's lemma to get $$W_t^{2k} = \frac{1}{2} 2k(2k-1) \int_0^t W_s^{2k-2}ds + 2k\int_0^t W_s^{2k-1}dW_s$$ Taking expectations, using the fact that the second integral is a martingale, and using Fubini's theorem, you get $$E(W_t^{2k}) = \frac{1}{2}2k(2k-1) \int_0^tE(W_s^{2k-2})ds$$ You can proceed inductively at this step, but I leave it to you to show that $$E(W_t^{2k}) = \frac{(2k)!}{2^k k!} t^k$$