= Solution
{c}
Use the auxiliary diffusion $dX_v=\mu X_v\,dv+\sigma X_v\,dW_v$, started at $X_t=x$. Under sufficient regularity and integrability for the <Feynman-Kac formula>,
$$
\boxed{F(t,x)=\mathbb E\left[
e^{-r(T-t)}\Psi(X_T)+
\int_t^Te^{-r(v-t)}d(v,X_v)\,dv
\,\middle|\,X_t=x\right].}
$$
To verify the signs, apply Itô's formula to $e^{-r(v-t)}F(v,X_v)$. Its drift is $e^{-r(v-t)}(F_t+\mu X_vF_x+\sigma^2X_v^2F_{xx}/2-rF)=-e^{-r(v-t)}d(v,X_v)$. Adding the accumulated source integral makes a martingale; taking expectations through $T$ proves the formula.
Here the auxiliary drift is the printed $\mu$, rather than automatically $r$: this part is a general terminal-value PDE, whose diffusion must match its generator.
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