I want to calculate $\log E[\exp(-\sqrt{d} S \epsilon)]$, where $\epsilon \sim N(0,1)$ and everything else is deterministic. The result should be $\frac{d}{2}||S||^2$ but why?
2026-04-12 13:28:48.1776000528
Expected Value of Exponential
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Hint: for $X\sim \mathcal{N}(\mu,\sigma)$ and $Y:=e^X$, then $Y$ is lognormal distributed with mean $E[Y]=e^{\mu+\frac{\sigma^2}{2}}$. If you don't know this result, it would be a good exercise.