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A self-financing portfolio is admissible when its wealth satisfies the prescribed lower-bound restriction, typically a deterministic bound below. Nonnegative wealth is sufficient. This condition permits supermartingale bounds on discounted gains and excludes doubling strategies.

A nonnegative supermartingale, or more generally a supermartingale whose negative parts have uniformly bounded expectations, converges almost surely to a finite integrable random variable.

For a square-integrable discounted claim martingale $dM=hdW$ and discounted stock $d\tilde{S}=\tilde{S}\sigma dW$, choose stock holdings $h/(\tilde{S}\sigma )$. The remaining wealth lies in the bond. A nonnegative claim makes this replication admissible; discounted admissible wealth is a supermartingale, establishing the minimal initial cost.

Let $\nu =\sigma \sqrt{(1-e^{-2r\tau })/(2r)}$ and $d=(s-K{e}^{-r\tau })/\nu$, where $\tau =T-t$. The displayed discounted Gaussian positive-part expectation prices a call in the arithmetic stock model with constant volatility. Its delta is $\Phi (d)$. The stock holding $C_s$ and bank-account holding $(C-s{C}_s)/B$ replicate the payoff with nonnegative wealth. A nonnegative discounted-wealth supermartingale bound proves this is the least replication capital.

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.

For a zero-starting continuous local martingale $X$, the stochastic exponential satisfiesThe Holder inequality and the supermartingale property of the first exponential bound its stopped $p$th moment by $(\mathbb{E}{e}^{C(p,q)X_T}{)}^{(q-1)/q}$. The subtraction in the coefficient is $\sqrt{pq}-1$, not a square root of $pq-1$.

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 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.

Every nonnegative local martingale is a supermartingale. In particular, one that starts from zero is identically zero: its nonnegative value at every time has expectation at most zero.