= Solution
{c}
Assume the SDE has the required solution and $f$ is sufficiently smooth. <Itô's formula> along that solution gives
$$
df(u,X_u)=
\left(f_t+\mu f_x+\tfrac12\sigma^2f_{xx}\right)(u,X_u)du
+\sigma(u,X_u)f_x(u,X_u)dW_u.
$$
The PDE cancels the first term. Integrating from $t$ to $T$ gives
$$
\Psi(X_T)-f(t,X_t)
=\int_t^T\sigma(u,X_u)f_x(u,X_u)dW_u.
$$
If 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 yields
$$
\boxed{f(t,x)=\mathbb E_{t,x}\Psi(X_T).}
$$
The 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.
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