FeynmanKacAnsatz

FeynmanKacAnsatz, commonly known as the Feynman–Kac formula, is a probabilistic representation that expresses the solution of a class of linear parabolic partial differential equations with a potential term as an expectation over diffusion processes. It provides a bridge between PDE theory and stochastic processes, enabling analytical insight and numerical methods based on stochastic calculus.

In its standard form, let X_t be a diffusion in R^d driven by dX_t = b(X_t) dt + σ(X_t) dW_t, with generator L u = ∑ b_i ∂_i u + 1/2 ∑ a_{ij} ∂_{ij} u and a = σ σ^T. For the backward equation ∂_t u + L u - c(x) u + f(x) = 0 on a time interval [0,T], with terminal condition u(T,x) = g(x), the Feynman–Kac formula provides the solution as a conditional expectation:

u(t,x) = E[ exp(-∫_t^T c(X_s) ds) g(X_T) + ∫_t^T exp(-∫_t^s c(X_r) dr) f(X_s) ds | X_t = x ].

This representation yields both the solution to the PDE and a practical means of computation, notably via Monte Carlo simulation.

Historical notes indicate the formula was developed through ideas attributed to physicist Richard Feynman and mathematician Mark Kac, who connected quantum and heat-type equations with stochastic processes. Variants exist for time-dependent coefficients, boundary conditions, and more general linear or semi-linear problems. In finance, the Feynman–Kac framework underpins risk-neutral valuation and option pricing, while in physics and engineering it facilitates analytical and numerical treatment of diffusion-type phenomena.