= Brownian moment recursion
{c}
{title2=$\beta_k(t)=\frac{k(k-1)}2\int_0^t\beta_{k-2}(s)\,ds$}
For standard one-dimensional Brownian motion, the Itô formula for $B^k$ gives this recursion. Stop at exits from $[-n,n]$ to make the stochastic integral a true martingale. Gaussian tail integrability and the <Doob Lp maximal inequality> dominate all stopped powers on a fixed horizon, justifying the limit. Odd moments vanish and $\mathbb EB_t^{2k}=(2k)!t^k/(2^kk!)$.
// Target: probability-and-statistics.bigb
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