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The Feynman-Kac Theorem: Linking PDEs to Expectations

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The Formal Theorem

Let Xt X_t be an Ito diffusion process defined by the Stochastic Differential Equation dXt=μ(Xt,t)dt+σ(Xt,t)dWt dX_t = \mu(X_t, t)dt + \sigma(X_t, t)dW_t . Consider the terminal value problem for a function u(x,t) u(x, t) defined on Rn×[0,T] \mathbb{R}^n \times [0, T] :
∂u∂t+μ(x,t)⋅∇u+12tr(σ(x,t)σ(x,t)T∇2u)−r(x,t)u+f(x,t)=0 \frac{\partial u}{\partial t} + \mu(x, t) \cdot \nabla u + \frac{1}{2} \text{tr}(\sigma(x, t)\sigma(x, t)^T \nabla^2 u) - r(x, t)u + f(x, t) = 0
subject to the terminal condition u(x,T)=Ψ(x) u(x, T) = \Psi(x) . Under suitable regularity conditions, the solution u(x,t) u(x, t) is given by the conditional expectation:
u(x,t)=EQ[∫tTe−∫tsr(Xτ,τ)dτf(Xs,s)ds+e−∫tTr(Xτ,τ)dτΨ(XT)∣Xt=x] u(x, t) = E^{Q} \left[ \int_{t}^{T} e^{-\int_{t}^{s} r(X_{\tau}, \tau) d\tau} f(X_s, s) ds + e^{-\int_{t}^{T} r(X_{\tau}, \tau) d\tau} \Psi(X_T) \mid X_t = x \right]

Analytical Intuition.

Imagine you are standing in a vast, foggy field (the state space) at time t t . You are moving according to a random, jittery path described by the diffusion Xt X_t . The Feynman-Kac theorem acts as a bridge between two worlds: the deterministic world of Partial Differential Equations (PDEs) and the probabilistic world of Stochastic Processes. The PDE tracks the 'evolution of information'—how a value flows backward from a terminal payoff Ψ(x) \Psi(x) to the present. The expectation, conversely, asks you to simulate every possible future path you could take until time T T , collect the discounted payoffs Ψ(XT) \Psi(X_T) and intermediate rewards f(Xs,s) f(X_s, s) , and average them. The theorem asserts that the solution to the complex PDE is exactly this average. It tells us that what we solve analytically as a geometric flow of a surface, we can estimate computationally by observing the collective history of countless particles diffusing through space. You are no longer just solving a static equation; you are averaging the destinies of all possible futures.
CAUTION

Institutional Warning.

Students often struggle to see why the PDE evolves backward in time. Remember that the expectation is conditioned on the state at time t t . As t t approaches T T , the time remaining to accrue payoff shrinks, forcing the PDE solution to converge to the terminal condition Ψ(x) \Psi(x) .

Academic Inquiries.

01

Why is the term r(x,t)u r(x, t)u present in the PDE?

The term r(x,t)u r(x, t)u represents the discounting factor. In financial contexts, it acts as a continuous interest rate, adjusting the future value back to its present value.

02

Does this theorem require the diffusion to be a Markov process?

Yes. The Feynman-Kac theorem relies on the Markov property of the underlying Ito diffusion, which ensures that the future evolution depends only on the current state Xt X_t , not the history.

Standardized References.

  • Definitive Institutional SourceØksendal, B., Stochastic Differential Equations: An Introduction with Applications.

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Institutional Citation

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NICEFA Visual Mathematics. (2026). The Feynman-Kac Theorem: Linking PDEs to Expectations: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/advanced-stochastic-processes/the-feynman-kac-theorem--linking-pdes-to-expectations

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