Formula relating stochastic processes to partial differential equations
The Feynman–Kac formula named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and stochastic processes. When Mark Kac and Richard Feynman were both on Cornell faculty, Kac attended a lecture 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.
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
^See Pham, Huyên (2009). Continuous-time stochastic control and optimisation with financial applications. Springer-Verlag. ISBN978-3-642-10044-4.
^Kac, Mark (1949). "On Distributions of Certain Wiener Functionals". Transactions of the American Mathematical Society. 65 (1): 1–13. doi:10.2307/1990512. JSTOR1990512. This paper is reprinted in Baclawski, K.; Donsker, M. D., eds. (1979). Mark Kac: Probability, Number Theory, and Statistical Physics, Selected Papers. Cambridge, Massachusetts: The MIT Press. pp. 268–280. ISBN0-262-11067-9.
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