Consider the two Brownian motions $W(t_{1})$ and $W(t_{2})$ with $(t_{2}\geq t_{1})$. We are interested in the following process:
$\mathbb{E}[e^{W(t_{2})-W(t_{1})}]$
We know $W(t_{2})-W(t_{1})$ is distributed as $N(0,t_{2}-t_{1})$ can we hence rewrite this expectation as:
$\int_{-\infty }^{\infty }xe^{{\sqrt{t_{2}-t_{1}}}x}f(x)dx$
(where is the density of a standard normal random variable)?