Past exam of the probability, measure and martingales course of the University of Oxford / 2023 / 1 / c / i / Solution

The first entrance time ${\tau }_K$ is a stopping time, because $\{{\tau }_K\le n\}={\bigcup }_{j=0}^n\{M_j\le -K\}\in {\mathcal{F}}_n$. Before this time $M_n>-K$; at entrance, the bounded increments give $M_{{\tau }_K}>-K-L$. Thus the stopped martingaleis a nonnegative martingale. Here the stopped process is a martingale since its increment is $1_{\{{\tau }_K>n\}}(M_{n+1}-M_n)$, whose conditional expectation given ${\mathcal{F}}_n$ is zero.

The martingale convergence theorem states that a nonnegative martingale converges almost surely to a finite integrable random variable; indeed its expectations are constant, giving the required bound in the L1 norm. Apply this theorem to $(N_n)$. On $\{{\tau }_K=\infty \}$ the stopped process equals the original one at every time. HenceThe overshoot bound is why the assumption of bounded increments matters.