Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 202 / 5 / 3

With $A_t={\int }_0^t\mu (s)ds$, the product rule givesso $X_t{e}^{-A_t}$ is a local martingale under $\mathbb{P}$.

Set $\theta (t)=\mu (t)/\sigma (t)$. This is bounded and compactly supported, so Novikov condition holds anddefines a probability measure $d\mathbb{Q}=Z_{\infty }d\mathbb{P}$. By the Girsanov theorem, $W_t=B_t+{\int }_0^t\theta (s)ds$ is Brownian under $\mathbb{Q}$, andThus $X$ is a local martingale under $\mathbb{Q}$.