The Feynman–Kac formula, named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations and stochastic processes. In 1947, when Kac and Feynman were both faculty members at Cornell University, Kac attended a presentation of Feynman's and remarked that the two of them were working on the same thing from different directions. The Feynman–Kac formula resulted, which proves rigorously the real-valued case of Feynman's path integrals. The complex case, which occurs when a particle's spin is included, is still an open question.

It offers a method of solving certain partial differential equations by simulating random paths of a stochastic process. Conversely, an important class of expectations of random processes can be computed by deterministic methods.

Consider the partial differential equation $${\displaystyle {\frac {\partial }{\partial t}}u(x,t)+\mu (x,t){\frac {\partial }{\partial x}}u(x,t)+{\tfrac {1}{2}}\sigma ^{2}(x,t){\frac {\partial ^{2}}{\partial x^{2}}}u(x,t)-V(x,t)u(x,t)+f(x,t)=0,}$$ defined for all ${\displaystyle x\in \mathbb {R} }$ and ${\displaystyle t\in [0,T]}$, subject to the terminal condition $${\displaystyle u(x,T)=\psi (x),}$$ where ${\displaystyle \mu ,\sigma ,\psi ,V,f}$ are known functions, ${\displaystyle T}$ is a parameter, and ${\displaystyle u:\mathbb {R} \times [0,T]\to \mathbb {R} }$ is the unknown. Then the Feynman–Kac formula expresses ${\displaystyle u(x,t)}$ as a conditional expectation under the probability measure ${\displaystyle Q}$

${\displaystyle u(x,t)=\operatorname {E} ^{Q}\left[g_{\tau }(t,T)\,\psi (X_{T})+\int _{t}^{T}g_{s}(t,\tau )\,f(X_{\tau },\tau )\,d\tau \,{\Bigg |}\,X_{t}=x\right]}$

where ${\displaystyle X}$ is an Itô process satisfying $${\displaystyle dX_{t}=\mu (X_{t},t)\,dt+\sigma (X_{t},t)\,dW_{t}^{Q},}$$ ${\displaystyle g_{\tau }}$ and ${\displaystyle g_{s}}$ are functions defined as $${\displaystyle g_{\kappa }(a,b)=\exp \left(-\int _{a}^{b}V(X_{\kappa },\kappa )\,d\kappa \right)}$$ where ${\displaystyle \kappa }$ can be substituted for ${\displaystyle \tau }$ or ${\displaystyle s}$ as appropriate, and ${\displaystyle W_{t}^{Q}}$ a Wiener process (also called Brownian motion) under ${\displaystyle Q}$.

Suppose that the position ${\displaystyle X_{t}}$ of a particle evolves according to the diffusion process $${\displaystyle dX_{t}=\mu (X_{t},t)\,dt+\sigma (X_{t},t)\,dW_{t}^{Q}.}$$ Let the particle incur "cost" at a rate of ${\displaystyle f(X_{s},s)}$ at location ${\displaystyle X_{s}}$ at time ${\displaystyle s}$. Let it incur a final cost at ${\displaystyle \psi (X_{T})}$.

Also, allow the particle to decay. If the particle is at location ${\displaystyle X_{s}}$ at time ${\displaystyle s}$, then it decays with rate ${\displaystyle V(X_{s},s)}$. After the particle has decayed, all future cost is zero.

Then ${\displaystyle u(x,t)}$ is the expected cost-to-go, if the particle starts at ${\displaystyle (t,X_{t}=x).}$