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

Write the diffusion and drift coefficients asBoth are continuously differentiable, withThese derivatives are bounded. For example $2|x|\le 1+x^2$ bounds both absolute values by one. The mean value theorem therefore makes both coefficients globally Lipschitz. They are also bounded, so satisfy the linear growth condition for an SDE.

The global existence theorem for stochastic differential equations with Lipschitz coefficients consequently givesIn particular, there is no finite explosion time. The solution has continuous paths, so all subsequent level-hitting and stopping arguments apply.

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.