Let $\mu,\sigma:[0,\infty)\to\mathbb{R}$ be deterministic continuous functions, assume that $\sigma$ is bounded below by a strictly positive constant and that $\mu$ has compact support. Suppose that $X$ is a solution to the SDE $$dX_t=X_t(\mu(t)dt+\sigma(t)dB_t)$$ with $X_0=1$ with respect to the probability measure $\mathbb{P}$.
The first part of the question, which I have completed, was to show that $X_te^{-\int_0^t\mu(s)ds}$ is a local martingale under $\mathbb{P}$, which is a fairly simple consequence of Ito's formula. The second question asks me to find a probability measure $\mathbb{Q}$ under which $X_t$ is a local martingale. (I think) the only results regarding change of measure in my lecture course have been Girsanov's theorem, but I can't see this directly as an application of Girsanov's theorem, so I am stuck. Hints and advice would be greatly appreciated, thanks!