Past exam of the stochastic differential equations course of the University of Oxford / 2011 / 1 / c / Solution

Let $p=\mathbb{P}(y_{\tau }=\alpha )$. For $k\ne 2$, the bounded stopped martingale from part (b) converges to $y_{\tau }^{2-k}$, so the dominated convergence theorem givesTherefore the boundary hitting probability from a diffusion scale function isFor $k=2$, applying Itô's formula to $\log Y_t$ gives a bounded stopped martingale instead, yieldingIf $k>2$ and $\beta >1$ is fixed, the first expression tends to zero as $\alpha \downarrow 0$. A continuous path that reaches zero before it reaches $\beta$ must first reach every such $\alpha$ before $\beta$, so that event has probability zero. Any path hitting zero in finite time has a finite maximum up to that time, and therefore belongs to one of these zero-probability events with an integer upper boundary $\beta >1$. Their countable union still has probability zero. Thus the process does not hit zero almost surely. This is the relevant part of the Hitting-zero classification for a Bessel process.