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Write $D_k=M_k-M_{k-1}$ for the martingale difference sequence. Bounded increments mean that one deterministic $C<\infty$ satisfies $|D_k|\le C$ for every $k$, almost surely. Then $\mathbb{E}{D}_k=0$ and, for $j<k$,The increments are uncorrelated random variables, uniformly bounded in $L^2$. Applying the supplied strong law for uniformly L2-bounded uncorrelated random variables gives $n^{-1}{\sum }_{k=1}^n{D}_k\to 0$ almost surely. Also $M_0/n\to 0$, because $M_0$ is finite almost surely. Therefore a martingale with bounded increments has zero linear growth:This is the strong law for martingales with bounded increments.

For the submartingale, let $\Delta Y_k=Y_k-Y_{k-1}$ and define its predictable drift and compensator byThis is the Doob decomposition in discrete time. The process $N_n$ is a martingale. Since $|\Delta Y_k|\le C$, we have $0\le a_k\le C$, so $|N_k-N_{k-1}|\le 2C$. The result just proved gives $N_n/n\to 0$ almost surely, and $A_n\ge 0$. Consequently the strong law for submartingales with bounded increments isThe compensator may grow, so the conclusion is a lower bound rather than a claim of convergence to zero.

Definefor $t\ge 1$, with $A_0=M_0=0$. The supermartingale property makes $\Delta A_t\ge 0$, and it is ${\mathcal{F}}_{t-1}$-measurable, so $A$ is previsible and nondecreasing. The increments of $M$ have conditional mean zero, so $M$ is a martingale. Finally,and summation gives the Doob decomposition in discrete time

Let $U=U_0+M-A$ be its Doob decomposition in discrete time and choose the martingaleSince $Z_t\le U_t$ and $A_t\ge 0$,For the first optimal stopping time ${\tau }^{\ast }$, the complementarity for the Snell envelope compensator implies $A_{{\tau }^{\ast }}=0$: all compensator increments before ${\tau }^{\ast }$ vanish. Since $Z_{{\tau }^{\ast }}=U_{{\tau }^{\ast }}$,Thereforepathwise, and taking expectations proves the asserted equality.

If a submartingale has uniformly bounded increments, then ${\mathrm{liminf}}_n{Y}_n/n\ge 0$ almost surely. In its Doob decomposition in discrete time, the predictable compensator is nondecreasing, while the martingale part has bounded increments. The strong law for martingales with bounded increments makes the latter negligible on the linear scale.