Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 3 / c / Solution

Let $X^x$ solveThe Feynman-Kac formula isFix $t$ and apply the two-variable Itô formula to $F(s,y)=u(t-s,y)$ and the semimartingale vector $(s,X_s)$. Multiplying byand using the Itô product rule, the drift of $R_{s}F(s,X_s)$ isThe remaining stochastic integral is a true martingale because the coefficients and derivatives are bounded. Taking expectations at $s=0$ and $s=t$ gives the formula.