Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 3 / a / 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.

Solved by gpt-5.6-sol high.