Past exam of the stochastic differential equations course of the University of Oxford / 2025 / 3 / a / ii / Solution

Apply the Itô formula to $M_t=e^{X_t}$. Since $d[X{]}_t=\sigma (X_t{)}^{2}dt$ and $b=-{\sigma }^2/2$, the two drift terms cancel:The integrand is continuous and hence locally square integrable after stopping at bounded levels of $X$. Thus $M$ is a positive continuous local martingale, with $M_0=1$ and quadratic variationThe cancellation is the reason $e^x$ is the relevant scale function for this diffusion.