Question
Let $W_t$ be the standard Wiener process, what is $\mathbb{E}[e^{W_t}]$?
Solution
We will use the moment generating function (MGF) for $W_t\sim\mathcal{N}(0,t)$. The MGF for Gaussian is $M(s) = \exp\left(\mu s+ (1/2)\sigma^2s^2\right)$, plug in $\mu=0, \sigma=\sqrt{t}$, we have $M(t) = e^{(1/2)s^2}$ and $M(1) = \mathbb{E}[e^{W_t}] = \exp((1/2)t)$.