Expectation of exponential of Brownian motion

Question

I want to compute the following expectation:

$\mathbb{E}[\int_0^\infty-e^{-\mu t+\sigma W_t}dt]$

where $W_t$ is a brownian motion, $\mu$ and $\sigma$ constant.

I am already stuck at computing the integral. I don't know how to solve something like $\int_0^\infty e^{W_t} dt$ to begin with. Any help is appreciated.

Answer 1

Hint: Apply Tonelli's (or Fubini's) theorem and use that the exponential moments of Gaussian random variables can be calculated explicitly.