Supermartingale

An integrable adapted process $(V_n)$ is a supermartingale when $\mathbb{E}[V_{n+1}\mid {\mathcal{F}}_n]\le V_n$.

A local supermartingale becomes a supermartingale after stopping along an increasing sequence of stopping times tending to infinity. Localization allows an Itô formula comparison before global integrability is established. Additional lower bounds or uniform integrability are needed to pass from the local comparison to an unrestricted expectation inequality.

For a positive stock satisfying $dS=S\sigma dW$, the Itô formula gives $d\sqrt{S}=\frac{1}{2}\sqrt{S}\sigma dW-\frac{1}{8}\sqrt{S}{\sigma }^{2}dt$. Localization and the conditional Fatou lemma make this nonnegative local supermartingale a true supermartingale.

For a right-continuous nonnegative supermartingale $M$ and $R>0$, the first level-hit time $\sigma$ and the optional stopping theorem give $R\mathbb{P}(\sigma \le T)\le \mathbb{E}[M_{\sigma \wedge T}]\le \mathbb{E}[M_0]$. Let $T\to \infty$ and first use levels $R-\epsilon$, then $\epsilon \downarrow 0$, to include a supremum not attained. For a continuous nonnegative local martingale on a stochastic interval, apply the argument on increasing localized compact subintervals, then pass to their limit. Letting $R\to \infty$ shows that its running supremum is finite almost surely.

If a nonnegative supermartingale reaches zero at a discrete stopping time, its later nonnegative values have conditional expectation at most zero and must also be zero almost surely.

A discrete-time supermartingale $X$ is a martingale exactly when $\mathbb{E}[X_n]=\mathbb{E}[X_0]$ for every $n$. If expectations are constant, the nonnegative variablehas expectation zero and therefore vanishes almost surely.

If a supermartingale $V$ dominates an adapted reward process $Z$, optional sampling gives $\mathbb{E}{Z}_{\tau }\le V_0$ for every bounded stopping time $\tau$.