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

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.