Submartingale

Remarks

Let $S_n = \xi_1 + \dots + \xi_n$, where the $\xi_i$ are independent with $P(\xi_i = 1) = P(\xi_i = -1) = 1/2$, and let $\mathcal{F}_n = \sigma(\xi_1, \dots, \xi_n)$. Then $X_n = S_n^2$ is a submartingale: it is adapted, $\abs{X_n} \le n^2$, and $E[S_{n+1}^2 \mid \mathcal{F}_n] = S_n^2 + 2 S_n E[\xi_{n+1}] + E[\xi_{n+1}^2] = S_n^2 + 1 \ge S_n^2$. The process $-S_n^2$ is adapted and integrable but fails clause 3, since $E[-S_{n+1}^2 \mid \mathcal{F}_n] = -S_n^2 - 1$. In discrete time $T = \{0, 1, 2, \dots\}$, clause 3 is equivalent to $E[X_{n+1} \mid \mathcal{F}_n] \ge X_n$ for each $n$, by the tower property. A process is a martingale if and only if it is both a submartingale and a supermartingale. If $(M_t)$ is a martingale and $\varphi$ is convex with $E[\abs{\varphi(M_t)}] < \infty$ for all $t$, then $(\varphi(M_t))$ is a submartingale by the conditional form of Jensen's inequality. Taking expectations in clause 3 shows that $t \mapsto E[X_t]$ is non-decreasing.