Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 29K / a / Solution

For $t>0$, differentiateThe growth assumptions justify differentiation under the expectation, and Gaussian integration by parts givesSince $u(0)=f(0)$, integration from $0$ to $t$ yields the Brownian transition semigroup identityContinuity at zero covers $t=0$.