Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 6 / c / Solution

Put ${\mathcal{F}}_n=\sigma (X_0,\dots ,X_n)$ and $\tau =T_0\wedge T_y$. Before $\tau$, the integer-valued increment belongs to $\{-1,0,1\}$. The martingale property givesTheir sum is at least $1/2$, so each conditional probability is at least $1/4$. From any state in $\{1,\dots ,y-1\}$, a run of at most $y$ upward moves reaches $y$ and has conditional probability at least $4^{-y}$. Applied in successive blocks of $y$ steps, this givesso $\tau <\infty$ almost surely.

The stopped process $X_{n\wedge \tau }$ takes values in $[0,y]$, hence is a bounded martingale and has uniform integrability. The optional sampling theorem for a supermartingale givesbecause $X_{\tau }$ is zero or $y$. Therefore