Brownian moment recursion (${\beta }_k(t)=\frac{k(k-1)}{2}{\int }_0^t{\beta }_{k-2}(s)ds$)

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 $\mathbb{E}{B}_t^{2k}=(2k)!t^k/(2^{k}k!)$.

// Target: probability-and-statistics.bigb

The process $B_t^k-\mathbb{E}{B}_t^k$ is a martingale only for $k=1,2$. For higher powers its finite-variation drift is $k(k-1)(B_t^{k-2}-\mathbb{E}{B}_t^{k-2})/2$, which is not identically zero. Subtracting the random compensator $k(k-1){\int }_0^t{B}_s^{k-2}ds/2$ instead produces a true martingale on each finite horizon.

// Target: probability-and-statistics.bigb