Past exam of the stochastic differential equations course of the University of Oxford / 2025 / 1 / a / i / Solution

A convenient sufficient condition is that $f\in C^2({\mathbb{R}}^d)$ has a bounded gradient. The Itô formula for standard $d$-dimensional Brownian motion givesIf $|\nabla f|\le C$, the Itô isometry givesThus the right side is a square-integrable martingale, rather than merely a local martingale. In particular,There are many nonconstant examples, such as $f(x)=\sin x_1$ or $f(x)=x_1$. More generally it is enough that $\mathbb{E}{\int }_0^t|\nabla f(B_s){|}^{2}ds<\infty$ for each finite $t$. Smooth test functions with compact support satisfy the bounded-gradient condition and will provide the domain in the next part.