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 faculty members at Cornell University, 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-valued 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[edit]
Consider the partial differential equation
where is an Itô process satisfying
Intuitive interpretation[edit]
Suppose that the position of a particle evolves according to the diffusion process
Also, allow the particle to decay. If the particle is at location at time , then it decays with rate . After the particle has decayed, all future cost is zero.
Then is the expected cost-to-go, if the particle starts at
Partial proof[edit]
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
Since
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:
Remarks[edit]
- 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 .
- More succinctly, letting be the infinitesimal generator of the diffusion process,
- This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods.
- 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
Applications[edit]
Finance[edit]
In quantitative finance, the Feynman–Kac formula is used to efficiently calculate solutions to the Black–Scholes equation to price options on stocks[7] and zero-coupon bond prices in affine term structure models.
For example, consider a stock price undergoing geometric Brownian motion
Quantum mechanics[edit]
In quantum chemistry, it is used to solve the Schrödinger equation with the Pure Diffusion Monte Carlo method.[8]
See also[edit]
- Itô's lemma
- Kunita–Watanabe inequality
- Girsanov theorem
- Kolmogorov forward equation (also known as Fokker–Planck equation)
- Stochastic mechanics
References[edit]
- ^ Kac, Mark (1987). Enigmas of Chance: An Autobiography. University of California Press. pp. 115–16. ISBN 0-520-05986-7.
- ^ Glimm, James; Jaffe, Arthur (1987). Quantum Physics: A Functional Integral Point of View (2 ed.). New York, NY: Springer. pp. 43–44. doi:10.1007/978-1-4612-4728-9. ISBN 978-0-387-96476-8. Retrieved 13 April 2021.
- ^ Øksendal, Bernt (2003). "Theorem 3.2.1.(iii)". Stochastic Differential Equations. An Introduction with Applications (6th ed.). Springer-Verlag. p. 30. ISBN 3540047581.
- ^ "PDE for Finance".
- ^ See Pham, Huyên (2009). Continuous-time stochastic control and optimisation with financial applications. Springer-Verlag. ISBN 978-3-642-10044-4.
- ^ Kac, Mark (1949). "On Distributions of Certain Wiener Functionals". Transactions of the American Mathematical Society. 65 (1): 1–13. doi:10.2307/1990512. JSTOR 1990512. 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. ISBN 0-262-11067-9.
- ^ Paolo Brandimarte (6 June 2013). "Chapter 1. Motivation". Numerical Methods in Finance and Economics: A MATLAB-Based Introduction. John Wiley & Sons. ISBN 978-1-118-62557-6.
- ^ Caffarel, Michel; Claverie, Pierre (15 January 1988). "Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman–Kac formula. I. Formalism". The Journal of Chemical Physics. 88 (2): 1088–1099. Bibcode:1988JChPh..88.1088C. doi:10.1063/1.454227.
Further reading[edit]
- Simon, Barry (1979). Functional Integration and Quantum Physics. Academic Press.
- Hall, B. C. (2013). Quantum Theory for Mathematicians. Springer.