This is an old revision of this page, as edited by 2001:f70:8100:8400:b4ef:ef50:be4f:ef64(talk) at 04:45, 13 August 2023(→Theorem: Improve typesetting. I played with using \exp\left(...\right) as well, but found this the simplest form to ingest the formula. One big advantage is that the similarity between the two exponentials becomes blatant, at the price of making the upper limit appear in a very small font, but I guess the reader is primed to find that difference.). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.
Revision as of 04:45, 13 August 2023 by 2001:f70:8100:8400:b4ef:ef50:be4f:ef64(talk)(→Theorem: Improve typesetting. I played with using \exp\left(...\right) as well, but found this the simplest form to ingest the formula. One big advantage is that the similarity between the two exponentials becomes blatant, at the price of making the upper limit appear in a very small font, but I guess the reader is primed to find that difference.)
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. 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.[1] 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.[2]
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.
Theorem
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 expectation
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,[3] 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 [4] with modifications to account for .
The expectation formula above is also valid for N-dimensional Itô diffusions. The corresponding partial differential equation for becomes:[5] where, i.e. , where denotes the transpose of .
When originally published by Kac in 1949,[6] 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).