Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 29K / c / Solution

Let ${\mathcal{F}}_s=\sigma (W_r:0\le r\le s)$. For $0\le s<t$, the increment $W_t-W_s$ is independent of ${\mathcal{F}}_s$ and distributed as $\sqrt{t-s}Z$. Apply part (a) to the translated function $u\mapsto f(W_s+u)$, conditionally on ${\mathcal{F}}_s$:ThereforeThe assumed growth conditions supply integrability, so $(M_t)$ is the Brownian compensator martingale.