Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 35 / 4 / a

Let $X,Y$ be two solutions with the same Brownian driver and the same initial value. Write $D=X-Y$, and take a common Lipschitz constant $L$ for both coefficients. Stop when $|X|+|Y|$ reaches $n$. Before this stopping time ${\tau }_n$, all difference integrands below are bounded. The Itô formula givesThe stopped stochastic integral is a true martingale on every finite horizon. Lipschitz continuity therefore yieldsSince $D_0=0$, the Gronwall inequality makes the left side zero. Let $n\to \infty$; the continuous global solutions are bounded on each compact time interval, so their stopping times exhaust it. Equality first at every rational time and then by continuity gives indistinguishability. This proves pathwise uniqueness for globally Lipschitz drift and diffusion coefficients. The argument does not require the initial value to have a global second moment: after stopping, the difference is bounded and initially zero.