Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 202 / 4 / c

Let $X$ and $Y$ be two solutions with the same initial value and Brownian motion, and put $Z=X-Y$. Itô formula givesStop when either process or the stochastic integral becomes large. Taking expectations, using the assumed one-sided Lipschitz bound, and then removing the localization givesThe Gronwall inequality yields $\mathbb{E}{Z}_t^2=0$. Thus $X_t=Y_t$ almost surely at every rational time, and path continuity makes the two processes indistinguishable. This proves pathwise uniqueness.