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

Let ${\tau }_n$ localize $M$. The Itô isometry givesThus the stopped variables are bounded in $L^2$, and localization plus weak compactness shows that $M$ is a true martingale. The Itô formula applied to $f(x)=x^2$ givesAfter localization this is a martingale; the Burkholder-Davis-Gundy inequalities and $\mathbb{E}[M{]}_t<\infty$ supply the required local integrability, so it is a true martingale.

Part 1 gives $\mathbb{E}{M}_t^2=\mathbb{E}[M{]}_t$. Henceso $M$ is $L^2$-bounded.

The martingale product identity and the Itô isometry for cross terms giveSince $B_s\sim N(0,s)$, its moment-generating function gives $\mathbb{E}{e}^{B_s}=e^{s/2}$. Therefore the answer is

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.