Which powers of Brownian motion are martingales

Question

Let $W$ be a standard Brownian motion. Find every non-negative integer $n$ for which $W_t^n$ is a martingale, and report the sum of $n!$ over those values.

Short answer

$2$ . Ito gives a drift of $\tfrac12n(n-1)W_t^{n-2}$ , which is identically zero only for $n=0$ (a constant) and $n=1$ . The answer is $0!+1!=2$ .

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Lessons behind this question

The Four Faces of Brownian Motion
The Optional Stopping Theorem
Discrete-Time Martingales

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