Solution
= 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> states
$$
(b-a)\mathbb E U_N[a,b]\leq\mathbb E(M_N-a)^-.
$$
Equivalent conventions for a process with a prescribed initial holding add the corresponding endpoint term.