Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 40 / 4 / a

For $S_0>0$, the stochastic exponential solution stays strictly positive. Put $U_t=\sqrt{S_t}$. The Itô formula givesThus $U$ is a nonnegative local supermartingale. To justify the true supermartingale property, stop where $U$ or its stochastic integral exceeds successive bounds. The stopped Itô formula gives $\mathbb{E}[U_{T\wedge {\tau }_m}\mid {\mathcal{F}}_t]\le U_{t\wedge {\tau }_m}$ for $T\ge t$. The conditional Fatou lemma and ${\tau }_m\uparrow \infty$ yieldIn particular $\mathbb{E}{U}_T\le U_0<\infty$. The case $S_0=0$ is the identically zero process. This is the square-root stock supermartingale.