Optional sampling theorem for a supermartingale

For bounded stopping times $\sigma \le \tau$, a supermartingale satisfies $\mathbb{E}[V_{\tau }\mid {\mathcal{F}}_{\sigma }]\le V_{\sigma }$; equality holds for a martingale.

For an integrable supermartingale with the displayed increment bound, $|X_{T\wedge n}|\le |X_0|+KT$. Thus the stopped values are dominated by one integrable random variable, including $X_T$. Apply optional stopping at the bounded times $T\wedge n$ and then the dominated convergence theorem. For a martingale the same proof gives equality.

An adapted process with integrable coordinates is a martingale exactly when the displayed equality holds for every bounded stopping time. For the forward implication, expand $M_T-M_0={\sum }_{k=1}^N{1}_{\{T\ge k\}}(M_k-M_{k-1})$ and condition each term on ${\mathcal{F}}_{k-1}$. Conversely, for $A\in {\mathcal{F}}_n$ choose $T=n+1$ on $A$ and $T=n$ elsewhere. Subtract the equality for the constant stopping time $n$ to obtain $\mathbb{E}[1_A(M_{n+1}-M_n)]=0$, which is the defining conditional expectation identity.

If $e^{-\theta N_{t\wedge {\sigma }_K}}$ is a nonnegative supermartingale, with ${\sigma }_K$ the first time $N_t\le K$, the optional stopping theorem yields ${\mathbb{P}}_N({\sigma }_K<\infty )\le e^{-\theta (N-K)}$. Such a bound can prove escape of a backlog independently of heuristic fluid models.