Centered Brownian powers need not be martingales
= Centered Brownian powers need not be martingales
The process $B_t^k-\mathbb EB_t^k$ is a martingale only for $k=1,2$. For higher powers its finite-variation drift is $k(k-1)(B_t^{k-2}-\mathbb EB_t^{k-2})/2$, which is not identically zero. Subtracting the random compensator $k(k-1)\int_0^tB_s^{k-2}ds/2$ instead produces a true martingale on each finite horizon.
// Target: probability-and-statistics.bigb