Doob upcrossing inequality

If $(M_n)$ is a martingale and $U_N[a,b]$ counts its completed upcrossings of $[a,b]$ by time $N$, thenEquivalent conventions may add an initial endpoint term.

A discrete-time martingale with ${\sup }_n\mathbb{E}|M_n|<\infty$ converges almost surely to an integrable random variable. If the martingale is uniformly integrable, convergence also holds in $L^1$.

A right-continuous continuous-time martingale with ${\sup }_{t\ge 0}\mathbb{E}|M_t|<\infty$ has an almost-sure finite limit as $t\to \infty$.