I know that $\mathbb{E}[W(t)^2] = t$ and $\mathbb{E}[W(t)^4] = 3t^2$ for a standard Wiener process $W(t)$ when $t$ is a (non-stochastic) number. I also know that $\mathbb{E}[W(\tau)^2] = \mathbb{E}[\tau]$ for a stopping time $\tau$. The question is: can $\mathbb{E}[W(\tau)^4]$ be expressed in terms of $\mathbb{E}[\tau^n]$, too?
I would also appreciate if an accessible reference is introduced (at the level of Evans's "Introduction to Stochastic Differential Equations"). Also if there is a general result for higher moments of $W(\tau)$ that would be great too.
Best Answer
The process $W(t)^4-6tW(t)^2+3t^2$ is a martingale. So if $\tau$ is a reasonable stopping time (e.g., bounded) then $\Bbb E[W(\tau)^4] =6\Bbb E[\tau W(\tau)^2]-3\Bbb E[\tau^2]$. To go further you need to know something about the joint distribution of $\tau$ and $W(\tau)^2$.