The Feynman–Kac formula named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and stochastic processes. It offers a method of solving certain PDEs 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 PDE
defined for all x in R and t in [0, T], subject to the terminal condition
where μ, σ, ψ, V, f are known functions, T is a parameter and is the unknown. Then the Feynman–Kac formula tells us that the solution can be written as a conditional expectation
Let u(x, t) be the solution to above PDE. Applying Itō's lemma to the process
the third term is and can be dropped. We also have that
Applying Itō's lemma once again to , it follows that
The first term contains, in parentheses, the above PDE and is therefore zero. What remains is
Integrating this equation from t to T, one concludes that
Upon taking expectations, conditioned on Xt = x, and observing that the right side is an Itō integral, which has expectation zero, it follows that
The desired result is obtained by observing that
- The proof above is essentially that of  with modifications to account for .
- The expectation formula above is also valid for N-dimensional Itô diffusions. The corresponding PDE for becomes (see H. Pham book below):
- i.e. γ = σσ′, where σ′ denotes the transpose matrix of σ).
- When originally published by Kac in 1949, the Feynman–Kac formula was presented as a formula for determining the distribution of certain Wiener functionals. Suppose we wish to find the expected value of the function
- in the case where x(τ) is some realization of a diffusion process starting at x(0) = 0. The Feynman–Kac formula says that this expectation is equivalent to the integral of a solution to a diffusion equation. Specifically, under the conditions that ,
- where w(x, 0) = δ(x) and
- The Feynman–Kac formula can also be interpreted as a method for evaluating functional integrals of a certain form. If
- where the integral is taken over all random walks, then
- where w(x, t) is a solution to the parabolic partial differential equation
- with initial condition w(x, 0) = f(x).
- Itō's lemma
- Kunita–Watanabe theorem
- Girsanov theorem
- Kolmogorov forward equation (also known as Fokker–Planck equation)
- Simon, Barry (1979). Functional Integration and Quantum Physics. Academic Press.
- Hall, B. C. (2013). Quantum Theory for Mathematicians. Springer.
- Pham, Huyên (2009). Continuous-time stochastic control and optimisation with financial applications. Springer-Verlag.
- Kac, Mark (1949). "On Distributions of Certain Wiener Functionals". Transactions of the American Mathematical Society. 65 (1): 1–13. JSTOR 1990512. doi:10.2307/1990512.This paper is reprinted in Mark Kac: Probability, Number Theory, and Statistical Physics, Selected Papers, edited by K. Baclawski and M.D. Donsker, The MIT Press, Cambridge, Massachusetts, 1979, pp.268-280