the Feynman-Kac formula
/ FINE-mun KATS /
The Feynman-Kac formula is the dictionary that turns certain partial differential equations into expectations over diffusion paths. It says: to solve a parabolic or elliptic PDE whose differential operator is a diffusion generator, you can simply average the boundary/terminal data along the trajectories of the corresponding diffusion. This is the deepest expression of the probability-PDE link: a deterministic field equation is solved by Monte Carlo over random paths, and conversely an expectation over paths is computed by solving a PDE.
In the parabolic (terminal-value) form: if X solves dX = b(X) dt + sigma(X) dB with generator L, and we want u(t, x) solving partial_t u + L u - r(x) u + g(x) = 0 on [0, T) with terminal condition u(T, x) = h(x), then u(t, x) = E[ h(X_T) exp(- integral_t^T r(X_s) ds) + integral_t^T g(X_s) exp(- integral_t^s r(X_u) du) ds | X_t = x ]. The term r is a discount/killing rate (a potential) and g is a source. The proof idea is clean: apply Ito's formula to the process Y_s = u(s, X_s) exp(- integral_t^s r du) plus the accumulated source; the drift terms cancel exactly because u solves the PDE, leaving a martingale, and taking expectations between t and T gives the formula. The elliptic version represents the solution of L u - r u + g = 0 on a domain D with boundary data, using the diffusion run until it first exits D (the exit time tau): u(x) = E_x[ h(X_tau) exp(-integral_0^tau r ds) + integral_0^tau g(X_s) e^(...) ds ].
Feynman-Kac matters because it makes high-dimensional PDEs (the heat equation, the Black-Scholes equation, screened-Poisson and Schrodinger-type problems) computable by simulation, and it gives clean probabilistic proofs of existence and regularity. The honest caveats: the representation requires the PDE to be well-posed and the expectation to be finite — the killing term r is allowed to be negative only with care, since exp(- integral r) can blow up (this is the Kac potential / spectral subtlety, and is exactly why Schrodinger eigenvalue problems are delicate). One also needs integrability/growth conditions so the local martingale in the proof is a true martingale (a uniform-integrability or Novikov-type condition), and boundary regularity so that hitting the boundary is well-behaved. The classical Feynman path-integral is the formal analytic continuation of this honest probabilistic formula.
Black-Scholes pricing is Feynman-Kac with L the generator of geometric Brownian motion and r the constant interest rate: a European option worth h(S_T) at maturity has time-t price u(t, S) = E*[ e^(-r(T-t)) h(S_T) | S_t = S ] under the risk-neutral measure, and u solves partial_t u + r S partial_S u + (sigma^2/2) S^2 partial_SS u - r u = 0. The discount e^(-r(T-t)) is exactly the exp(-integral r) factor.
Black-Scholes as Feynman-Kac: the option price is a discounted expectation that solves a parabolic PDE.
The formula needs the local martingale in the proof to be a true martingale (an integrability/Novikov-type condition) and the expectation to be finite — a negative potential r can make exp(-integral r) blow up, which is exactly the subtlety in Schrodinger/spectral problems.