= Maximal inequality for a nonnegative supermartingale
{title2=$\mathbb P(\sup_t M_t\geq R)\leq\mathbb E[M_0]/R$}
For a right-continuous nonnegative <supermartingale> $M$ and $R>0$, the first level-hit time $\sigma$ and the <optional stopping theorem> give $R\mathbb P(\sigma\leq T)\leq\mathbb E[M_{\sigma\wedge T}]\leq\mathbb E[M_0]$. Let $T\to\infty$ and first use levels $R-\varepsilon$, then $\varepsilon\downarrow0$, 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.
Back to article page
