 Submartingale

A submartingale is an integrable adapted process satisfying $\mathbb{E}[X_{n+1}\mid {\mathcal{F}}_n]\ge X_n$ almost surely. Its conditional tendency is upward; an individual sample path can still fall. If $S_n$ is a mean-zero random walk with step variance ${\sigma }^2$, then $S_n^2$ is a submartingale because $\mathbb{E}[S_{n+1}^2\mid {\mathcal{F}}_n]=S_n^2+{\sigma }^2$. More generally a convex function of a martingale is a submartingale when integrable, by conditional Jensen inequality. Negating a submartingale gives a supermartingale.