Supermartingale

Remarks

Let $\xi_1, \xi_2, \dots$ be independent with $P(\xi_i = 1) = p$ and $P(\xi_i = -1) = 1 - p$, where $p \le 1/2$, and let $\mathcal{F}_n = \sigma(\xi_1, \dots, \xi_n)$. Then $S_n = \xi_1 + \dots + \xi_n$ is a supermartingale: it is adapted, $\abs{S_n} \le n$, and $E[S_{n+1} \mid \mathcal{F}_n] = S_n + (2p - 1) \le S_n$. For $p = 1/2$, the process $S_n^2$ is adapted and integrable but fails clause 3, since $E[S_{n+1}^2 \mid \mathcal{F}_n] = S_n^2 + 1$. $(X_t)$ is a supermartingale if and only if $(-X_t)$ is a submartingale. A process is a martingale if and only if it is both a submartingale and a supermartingale. Taking expectations in clause 3 shows that $t \mapsto E[X_t]$ is non-increasing.