$B_t^2$ 是鞅/次鞅/上鞅？

$B_{t}$ is a Brownian

Assume that $B_{t}$ is a Brownian motion. Determine whether $B_{t}^{2}$ is a martingale, a submartingale, or a supermartingale.

解析

对 $t>s$ ，写

B_t=B_s+(B_t-B_s),

其中增量与 $\mathcal{F}_s$ 独立且方差为 $t-s$ 。于是

\mathbb{E}[B_t^2\mid\mathcal{F}_s]=B_s^2+\mathbb{E}[(B_t-B_s)^2]=B_s^2+(t-s)\ge B_s^2.

所以 $B_t^2$ 是次鞅；并且

\boxed{B_t^2-t\ \text{是鞅}}.