The Feynman-Kac Formula

The Feynman-Kac formula connects partial differential equations (PDEs) to stochastic processes, expressing solutions of PDEs as expectations of functionals of stochastic processes.

Statement

Theorem7.1Feynman-Kac

Consider the PDE:

$\frac{\partial u}{\partial t} + \mu(x) \frac{\partial u}{\partial x} + \frac{1}{2}\sigma^2(x) \frac{\partial^2 u}{\partial x^2} - r(x) u + f(x) = 0$

with terminal condition $u(T, x) = g(x)$ . Let $X_t$ solve $dX_t = \mu(X_t) dt + \sigma(X_t) dB_t$ with $X_0 = x$ . Then:

$u(t, x) = \mathbb{E}\left[e^{-\int_t^T r(X_s) ds} g(X_T) + \int_t^T e^{-\int_t^s r(X_u) du} f(X_s) ds \,\Big|\, X_t = x\right].$

Intuition: The PDE solution is the expected discounted payoff under the stochastic dynamics. This is the foundation of derivative pricing.

ExampleHeat equation

For the heat equation $\frac{\partial u}{\partial t} = \frac{1}{2}\frac{\partial^2 u}{\partial x^2}$ with $u(T, x) = g(x)$ , Feynman-Kac gives:

$u(t, x) = \mathbb{E}[g(X_T) \mid X_t = x],$

where $X_t$ is Brownian motion. This probabilistic representation allows Monte Carlo methods for solving PDEs.

Summary

Feynman-Kac bridges PDEs and SDEs, expressing PDE solutions as expectations of path functionals. It is central to option pricing (Black-Scholes PDE) and numerical methods (Monte Carlo for PDEs).