Feynman-Kac Formula

Core Idea

The Feynman-Kac formula connects solutions of parabolic PDEs to expectations of stochastic processes. If u(x,t) solves ∂u/∂t + μ(x)∂u/∂x + (1/2)σ²(x)∂²u/∂x² - r(x)u = 0 with terminal condition u(x,T) = g(x), then u(x,t) = E[exp(-∫ₜᵀ r(X(s))ds) · g(X(T)) | X(t) = x], where dX = μ dt + σ dW. This provides a probabilistic representation of PDE solutions and a PDE representation of stochastic expectations — a deep bridge between analysis and probability.

Explainer

The Feynman-Kac formula is one of the deepest connections in mathematics: it equates solutions of partial differential equations with expectations of stochastic processes. The formula states that the solution of the parabolic PDE ∂u/∂t + μ(x)∂u/∂x + (1/2)σ²(x)∂²u/∂x² - r(x)u = 0 with terminal condition u(x,T) = g(x) can be represented as u(x,t) = E[exp(-∫ₜᵀ r(X(s))ds) · g(X(T)) | X(t) = x], where X follows the SDE dX = μ(X)dt + σ(X)dW. The PDE coefficients μ, σ determine the dynamics of the process; the terminal condition g determines the payoff; the coefficient r introduces discounting.

The proof in one direction (from stochastic representation to PDE) uses Itô's formula directly. Define u(x,t) = E[e^{-∫r ds} g(X(T)) | X(t) = x] and consider the process M(s) = e^{-∫ₜˢ r du} u(X(s), s) for s ∈ [t,T]. Apply Itô's formula to M: the drift must be zero because M is a martingale (it is a conditional expectation process). Setting the drift to zero yields exactly the PDE. The converse direction (from PDE solution to stochastic representation) reverses this argument: if u solves the PDE, then M(s) = e^{-∫ₜˢ r du} u(X(s), s) has zero drift by Itô's formula, hence is a martingale. Taking expectations at s = T gives u(x,t) = E[e^{-∫r ds} g(X(T)) | X(t) = x].

This bridge works in both directions practically. Given a PDE, you can solve it by simulating diffusion paths and averaging — this is the Monte Carlo method, which excels in high dimensions where grid-based PDE solvers fail catastrophically due to the curse of dimensionality. Conversely, given a stochastic expectation, you can write down the corresponding PDE and apply analytical techniques (separation of variables, Green's functions, transform methods) or efficient numerical PDE solvers. The choice depends on the problem structure: low-dimensional problems with smooth coefficients favor PDE methods; high-dimensional problems or problems with complex path-dependent payoffs favor Monte Carlo.

In mathematical finance, the Feynman-Kac formula is the link between the Black-Scholes PDE and risk-neutral pricing. The Black-Scholes PDE ∂V/∂t + rS(∂V/∂S) + (1/2)σ²S²(∂²V/∂S²) - rV = 0 is in exact Feynman-Kac form with the risk-neutral drift rS, diffusion σS, and discount rate r. The formula gives V(S,t) = E_Q[e^-r(T-t) payoff(S_T)], recovering the risk-neutral pricing formula. Richard Feynman arrived at this formula from the physics side (path integrals in quantum mechanics); Mark Kac gave the rigorous mathematical proof. The formula reveals that PDEs and expectations are two languages for the same mathematics — master both, and you can translate freely between them.