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

The Bessel process is stopped in the compact interval $[\alpha ,\beta ]$, where its drift and diffusion coefficients are bounded. In particular its stopped stochastic integral has expectation zero on every finite time interval.

For $k>1$, taking expectations in the stopped equation givesSince $Y_t\le \beta$, this bounds $\mathbb{E}(t\wedge \tau )$ uniformly in $t$. For $k<1$, the same argument with $1-\mathbb{E}{Y}_t$ gives $\mathbb{E}(t\wedge \tau )\le 2\beta (1-\alpha )/(1-k)$.

If $k=1$, the stopped process is a stopped Brownian motion. Itô's formula gives $\mathbb{E}(Y_t-1{)}^2=\mathbb{E}(t\wedge \tau )$, which is at most $\max \{(1-\alpha {)}^2,(\beta -1{)}^2\}$. Thus in every case the monotone convergence theorem gives $\mathbb{E}\tau <\infty$, so $\tau <\infty$ almost surely. By path continuity its value at exit is one of the two endpoints. They are disjoint possibilities, hence

For a twice continuously differentiable function, Itô's formula in this stopped diffusion readsFor $f(x)=x^{2-d}$,Setting $d=k$ eliminates the drift. Moreover $f^{\prime }$ is bounded on $[\alpha ,\beta ]$, so the stochastic integral is square-integrable on each finite horizon by the Itô isometry. Hence $f(Y_t)$ is a true martingale, not merely a local one. The process itself is bounded on the stopped interval. For $k=2$, this particular function is the constant one; the logarithmic scale function will supply the nontrivial hitting identity.

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.