Concept

Feynman–Kac formula

Summary

The Feynman–Kac formula, named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and stochastic processes. In 1947, when Kac and Feynman were both 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 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 defined for all and , subject to the terminal condition where are known functions, is a parameter, and is the unknown. Then the Feynman–Kac formula tells us that the solution can be written as a conditional expectationunder the probability measure such that is an Itô process driven by the equation with is a Wiener process (also called Brownian motion) under , and the initial condition for is . 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 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 to , one concludes that Upon taking expectations, conditioned on , 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 The proof above that a solution must have the given form is essentially that of with modifications to account for .

Official source

https://en.wikipedia.org/wiki/Feynman–Kac_formula

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