Supermartingale

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

If two supermartingales satisfy $X_N^1\ge X_N^2$ on the event that the stopping time $N$ is finite, the displayed pasted adapted process is a supermartingale. On $\{N>n\}$ its next value is at most $X_{n+1}^1$, because a possible switch at $n+1$ only decreases it. On $\{N\le n\}$ it follows the second supermartingale. Conditioning these inequalities proves the assertion. Finite repeated pastings inherit the same property, allowing a multiplicative trading construction for upcrossings.

An $L^1$-bounded supermartingale has an almost sure convergence limit $X_{\infty }$ in $L^1$. The conditional-expectation martingale $M_n=\mathbb{E}[X_{\infty }\mid {\mathcal{F}}_n]$ has uniform integrability and converges to $X_{\infty }$ both almost surely and in $L^1$. To identify its limit, test against every event in every finite-time sigma-algebra and use uniqueness on their generated sigma-algebra. Then $Y=X-M$ is a supermartingale tending to zero almost surely. There is no general nonnegativity conclusion for $Y$, nor general convergence in L1 to zero: the negative of the fair-coin doubling martingale is a counterexample to both stronger assertions.

If $U_t$ is a supermartingale and $\tau$ is a stopping time, then $U_{t\wedge \tau }$ is a supermartingale. Adaptedness follows from the stopping-time events, and integrability follows from $|U_{t\wedge \tau }|\le {\sum }_{j=0}^t|U_j|$. Its increment is $1_{\{\tau >t\}}(U_{t+1}-U_t)$. Since $\{\tau >t\}\in {\mathcal{F}}_t$, conditioning gives a nonpositive conditional mean. No integrability assumption on the possibly unbounded stopping time is required for this stopped-process assertion at each fixed finite $t$.

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 $\tau$.