What is $E[W_t ^2 e^{(\mu W_t - \frac{\sigma^2}{2}t)}]$?

Question

What is the expected value:

$E[W_t ^2 e^{(\mu W_t - \frac{\sigma^2}{2}t)}]$ where $W_t$ is a standard Brownian Motion and $\mu, \sigma >0$

One possible hint is: take $d/d \mu$ twice.

I don't know how to use it, could somebody help?

Answer 1

Hint: let $f(\mu)=Ee^{\mu W_t-\frac {\sigma^{2}t} 2}$. Then and $f''(\mu)=E W_t^{2}e^{\mu W_t-\frac {\sigma^{2}t} 2}$. Recall that $Ee^{\mu W_t}=e^{\mu^{2}t/2}$.