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 on Cornell faculty, 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 case of Feynman's path integrals. The complex case, which occurs when a particle's spin is included, is still unproven. 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.
A proof that the above formula is a solution of the differential equation is long, difficult and not presented here. It is however reasonably straightforward to show that, if a solution exists, it must have the above form. The proof of that lesser result is as follows. Let u be the solution to the above partial differential equation. Applying the product rule for Itô processes to the process one gets Since the third term is and can be dropped. We also have that Applying Itô's lemma to, it follows that The first term contains, in parentheses, the above partial differential equation 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 Itô integral, which has expectation zero, it follows that The desired result is obtained by observing that and finally
Remarks
The proof above that a solution must have the given form is essentially that of with modifications to account for.
The expectation formula above is also valid for N-dimensional Itô diffusions. The corresponding partial differential equation for becomes:
This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods.
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
