Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 25 / 4 / b / Solution

Write $f(z)=1/\cosh z$. Differentiating givesTherefore the diffusion operator satisfies $\frac{1}{2}{f}^{\prime \prime }+\tanh z{f}^{\prime }=-\frac{1}{2}f$. Applying the Itô formula to $Y_t=e^{t/2}f(X_t)$ cancels its drift:So $Y$ is a positive local martingale. On every finite interval $[0,R]$, $0<Y_t\le e^{R/2}$. The bounded local martingale criterion makes it a true martingale on that interval. Since $R$ is arbitrary,