Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 5 / c / Solution

LetDifferentiating and using the scale equation givesSince $b$ is bounded, $h$ has global Lipschitz continuity. Thushas a pathwise unique strong solution by the standard Lipschitz existence-and-uniqueness theorem for a stochastic differential equation. Applying the deterministic inverse $g^{-1}$ gives a strong solution $X$, and uniqueness of $Y$ gives pathwise uniqueness of $X$.