 Solution

Let $(M_k,{\mathcal{F}}_k{)}_{k=0}^N$ be a martingale, let $a<b$, and let $U_N[a,b]$ count the completed upcrossings of $[a,b]$ by time $N$. Doob upcrossing inequality statesEquivalent conventions for a process with a prescribed initial holding add the corresponding endpoint term.