Derivation — Feynman-Kac via Itô's Formula

The Feynman-Kac theorem states that $\varphi(t,x) = \mathbb{E}[e^{-\int_t^T r(s,X_s)ds} F(X_T) \mid X_t = x]$ solves a parabolic PDE. This derivation note carries out the argument in full: we apply Itô's formula to $e^{-\int_t^s r(u,X_u) \, du} \varphi(s, X_s)$, observe that the result must be a martingale (by the tower property), and conclude that the drift term must vanish — which is exactly the PDE. Evaluating at $s = T$ recovers the terminal condition.

The same argument yields the generator form of the PDE as a byproduct. Two related approaches — the martingale method and the tower-property method — are compared.