Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 5 / b / Solution

Since $V^{\prime }=v$, the fundamental theorem of calculus givesApplying the Itô formula before the lifetime, the two drift terms cancel:Thus $\psi (X)$ is a continuous local martingale. The increasing function $\psi$ is the scale function of a one-dimensional diffusion.