Past exam of the continuous martingales and stochastic calculus course of the University of Oxford / 2023 / 3 / a / ii / Solution

Let $X=(X^1,\dots ,X^d)$ be a continuous semimartingale, with $X^j=X_0^j+M^j+A^j$, where $M^j$ is a zero-starting continuous local martingale and $A^j$ is a continuous finite-variation process starting at zero. For $f\in C^{1,2}([0,\infty )\times {\mathbb{R}}^d)$, the Itô formula isHere $[M^j,M^k]$ is the quadratic covariation; the finite-variation process parts contribute no such term. The integrals against the finite-variation processes are understood pathwise, and those against the continuous local martingales are Itô integrals after localization.

For $d$-dimensional Brownian motion with independent standard coordinates, $d[B^j,B^k{]}_s={\delta }_{jk}ds$. In particular, if $f\in C^2$ is independent of time,where $\Delta$ is the Laplace operator. The same formulas hold up to exit from an open domain on which $f$ is smooth.