Using Ito's formula to calculate higher moments of the Wiener process?

Question

We have the $q$-th moment of the one-dimensional Wiener process at each time $t$:

$m_q(t)=E((W(t,.))^q)$, $q=1,2,...,t \geq 0$.

I am trying to prove that

$m_q(t)=\frac{1}{2}q(q-1)\left(\int_{0}^{t} m_{q-2}(s)ds\right)$ for all $q \geq 2$ and $t \geq 0$.

Is there an elegant way of showing this?

Answer 1

There is indeed an elegant way to prove this. To that effect, I give you the following two hints.

Hint 1: Find suitable $g(x)$ to apply Itô's formula to.

Edit Let $g(x) = x^q$ then $dg(x) = \frac{1}{2}q(q-1)x^{q-2}dt + qx^{q-1}dW_t$ or in integral form $$g(x) = \frac{1}{2}\int_0^tq(q-1)x^{q-2}ds + \int_0^tqx^{q-1}dW_s$$

Hint 2: Consider the behavior of Itô integrals under expectation.

EDIT Evaluate $$\mathbb{E}[g(x)] = \mathbb{E}[\frac{1}{2}\int_0^tq(q-1)x^{q-2}ds + \int_0^tqx^{q-1}dW_s]$$

If you need further help comment below your difficulties.