Write the $d$-dimensional continuous semimartingale as $X=X_0+M+A$, where $M$ is a continuous local martingale and $A$ is a continuous <finite-variation process>. For $f\in C^2(\mathbb R^d)$, <Itô formula> states
$$
f(X_t)=f(X_0)+\sum_{i=1}^d\int_0^t\partial_if(X_s)\,dX_s
+\frac12\sum_{i,j=1}^d\int_0^t\partial_{ij}f(X_s)\,d[X^i,X^j]_s.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 202 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-202/3/a/solution