 Solution

Assume the SDE has the required solution and $f$ is sufficiently smooth. Itô's formula along that solution givesThe PDE cancels the first term. Integrating from $t$ to $T$ givesIf this integral is a true martingale, for example if its integrand has finite expected integrated square, its conditional expectation at time $t$ is zero. For the diffusion started at $x$ at time $t$, this yieldsThe notation in the question is the same Markov conditional expectation. The martingale or corresponding uniform-integrability growth hypothesis is part of the usual Feynman-Kac formula justification: a local martingale alone does not automatically have zero expected increment. Standard smooth solutions with the customary integrability bounds satisfy this requirement.