 Feynman–Kac formula

The Feynman-Kac formula connects a partial differential equation to an expected value over paths of a diffusion process. Let $X$ be a nonexplosive Itô diffusion with infinitesimal generator $L$, let $V,g$ be bounded with $V\ge 0$, and let $u$ be a bounded classical solution ofThenThe Itô formula and Itô product rule make $e^{-{\int }_0^{s}V(X_r)dr}u(t-s,X_s)$ a local martingale; boundedness makes it a true martingale on the finite horizon. Its equal endpoint expectations prove the identity and uniqueness within the bounded solution class.

The coefficient $V$ acts as a position-dependent killing rate. For constant $V=\lambda$ and $g=1$, the formula gives $u(t,x)=e^{-\lambda t}$. For $V=0$ and Brownian motion, it gives the heat equation solution by averaging $g$ against the Brownian transition density. Sources, terminal data and boundary stopping lead to the parabolic Feynman-Kac formula with a source and elliptic Feynman-Kac formula.

The Feynman-Kac theorem is a fundamental result in stochastic processes, particularly in the context of linking partial differential equations (PDEs) with stochastic processes, specifically Brownian motion. It provides a way to express the solution of a certain type of PDE in terms of expectations of functionals of stochastic processes, such as those arising from Brownian motion.