The Feynman-Kac Formula: PDEs Solved by Random Paths

1. The precise statement

Fix a diffusion X on ℝⁿ solving the SDE dX_s = b(X_s) ds + σ(X_s) dW_s, where W is standard Brownian motion, and assume b, σ are Lipschitz with linear growth (so a unique strong solution exists and does not explode). Let L = ½ ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼ + ∑ᵢ bᵢ ∂ᵢ be its generator, with a = σσᵀ. Let V (the potential) be continuous and bounded below, and f continuous of at most polynomial growth.

Suppose u ∈ C¹,²([0,T]×ℝⁿ) solves the terminal-value problem

∂u/∂t + Lu − V·u = 0, u(T, x) = f(x),

with u of polynomial growth. Then u admits the stochastic representation

u(t, x) = 𝔼[ f(X_T) · exp(−∫ₜᵀ V(X_s) ds) | X_t = x ].

The conditioning X_t = x means we start the diffusion at x at time t. This is a uniqueness statement: any classical solution of the given growth must equal this expectation, so there is at most one.

2. The picture: random walkers carrying a discount

Imagine dropping a cloud of particles at x at time t, each executing the diffusion X. By time T they have spread out; a particle lands wherever its random path took it, and there we read off the payoff f(X_T). The solution at (t, x) is the average payoff over the cloud — this is already the Kolmogorov backward equation when V ≡ 0.

The potential V adds bookkeeping. Along each path we accumulate ∫ₜᵀ V(X_s) ds and weight the payoff by exp(−that). If you interpret V as a killing rate, a particle in a region of high V is more likely to be annihilated; exp(−∫V) is exactly the survival probability of a particle killed at rate V(X_s). So Feynman-Kac says: propagate, but let paths through high-potential regions decay. Positive V discounts; negative V amplifies. This is why the same formula prices options (V = interest rate) and computes quantum-mechanical ground states in imaginary time (V = the physical potential).

3. Key idea of the proof: build a martingale

The engine is Itô's formula plus the observation that a cleverly weighted process is a martingale. Define the discount factor D_s = exp(−∫ₜˢ V(X_r) dr) and consider

M_s = D_s · u(s, X_s), t ≤ s ≤ T.

Apply Itô to u(s, X_s) and the product rule with D_s (which has finite variation, dD_s = −V(X_s)D_s ds). The drift terms collect into

D_s (∂u/∂t + Lu − V·u)(s, X_s) ds = 0,

which vanishes precisely because u solves the PDE. What remains is the stochastic integral D_s (σᵀ∇u)(s,X_s)·dW_s, a local martingale. The polynomial-growth / boundedness hypotheses promote it to a true martingale (uniformly integrable on [t,T]). Then 𝔼[M_T | X_t=x] = M_t, i.e. u(t,x) = 𝔼[D_T f(X_T)], since M_t = u(t,x) and M_T = D_T u(T,X_T) = D_T f(X_T). The PDE's job is to kill the drift; the martingale property does the rest.

4. Canonical example: the heat equation and Black-Scholes

Heat equation. Take n = 1, b = 0, σ = 1, V = 0, so X_s = x + (W_s − W_t) is Brownian motion. The PDE ∂u/∂t + ½∂²u/∂x² = 0 with u(T,·)=f gives u(t,x) = 𝔼[f(x + W_{T−t})] = ∫ f(y) (2π(T−t))^{−1/2} exp(−(y−x)²/(2(T−t))) dy — the Gaussian heat kernel drops out of the expectation automatically.

Black-Scholes. Under the risk-neutral measure a stock follows dS = rS dt + σS dW. A European claim with payoff f(S_T) has price satisfying ∂V/∂t + ½σ²S²∂²V/∂S² + rS ∂V/∂S − rV = 0. Here the potential is the constant r, so Feynman-Kac gives the famous discounted-expectation formula

V(t, S) = e^{−r(T−t)} 𝔼[ f(S_T) | S_t = S ].

Plugging f = (S_T − K)⁺ and the lognormal law of S_T recovers the closed-form Black-Scholes price. The 'discount factor' e^{−r(T−t)} is exactly exp(−∫ₜᵀ r ds).

5. Why the hypotheses matter

V bounded below. If V can dive to −∞, the weight exp(−∫V) can blow up and the expectation diverge; the representation fails or gives +∞. Bounded-below V keeps exp(−∫ₜᵀV ds) ≤ e^{−(T−t)inf V} < ∞, so integrability survives.

Growth control on u and f. Without it, M_s is only a local martingale, and 𝔼[M_T] ≠ M_t can fail — the classic gap between local and true martingales. Polynomial growth plus the SDE's moment bounds (𝔼[sup|X_s|^p] < ∞) upgrades it. Drop this and you can have two distinct solutions, so the representation no longer pins down u.

Non-explosion / Lipschitz coefficients. Needed so X exists globally on [t,T]; superlinear drift can send X to ∞ in finite time, invalidating both Itô and the expectation.

Connections. Feynman-Kac is the parabolic cousin of the Kolmogorov backward equation, the Dirichlet-problem representation u(x)=𝔼[f(X_τ)] for harmonic functions (V=0, elliptic), and — via Wick rotation t↦it — Feynman's original path integral for the Schrödinger equation.

6. Applications and significance

Feynman-Kac is the bridge that made Monte Carlo methods for PDEs possible: to estimate u(t,x), simulate many paths of X, average f(X_T)exp(−∫V). In n dozens of dimensions this beats grid methods, which suffer the curse of dimensionality — the reason quantitative finance prices high-dimensional basket options by simulation.

In mathematical finance it underlies the entire risk-neutral pricing framework: every derivative price is a discounted expectation, and the pricing PDE and the expectation are Feynman-Kac duals. In quantum physics and statistical mechanics, imaginary-time Feynman-Kac represents e^{−tH} (with H = −½Δ + V) as a path integral, powering diffusion Monte Carlo and rigorous spectral bounds; Kac used it to study Wiener-sausage and ground-state energies. It also gives probabilistic proofs of PDE facts — maximum principles, Harnack inequalities, spectral gap estimates — and, run in reverse, lets PDE regularity theory certify smoothness of expectations. The formula is a cornerstone of modern stochastic analysis precisely because it converts an analytic problem into a sampling problem and back.

The PDE-probability dictionary: each analytic object corresponds to a probabilistic one.

PDE side	Probabilistic side	Role
Generator ½σ²Δ + b·∇	Diffusion X_t (SDE dX = b dt + σ dW)	Second-order operator ↔ Markov process
Potential term −V·u	Weight exp(−∫V(X_s)ds)	Zeroth-order coefficient ↔ path killing rate
Terminal data u(T,·)=f	Payoff f(X_T)	Boundary condition ↔ final observable
Source term +g	Additive ∫ₜᵀ g(X_s)e^{−∫V} ds	Inhomogeneity ↔ running cost
Elliptic −Lu + Vu = h	𝔼[∫₀^τ h(X_s)e^{−∫V}ds]	Stationary problem ↔ stopped expectation