 Solution

Use the auxiliary diffusion $d{X}_v=\mu X_{v}dv+\sigma X_{v}d{W}_v$, started at $X_t=x$. Under sufficient regularity and integrability for the Feynman-Kac formula,To verify the signs, apply Itô's formula to $e^{-r(v-t)}F(v,X_v)$. Its drift is $e^{-r(v-t)}(F_t+\mu X_v{F}_x+{\sigma }^2{X}_v^2{F}_{xx}/2-rF)=-e^{-r(v-t)}d(v,X_v)$. 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.