Feynman–Kac formula

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The Feynman–Kac formula named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and stochastic processes. It offers a method of solving certain PDEs 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 PDE

defined for all x in R and t in [0, T], subject to the terminal condition

where μ, σ, ψ, V, f are known functions, T is a parameter and is the unknown. Then the Feynman–Kac formula tells us that the solution can be written as a conditional expectation

under the probability measure Q such that X is an Itō process driven by the equation

with WQ(t) is a Wiener process (also called Brownian motion) under Q, and the initial condition for X(t) is X(t) = x.

Proof[edit]

Let u(x, t) be the solution to above PDE. Applying Itō's lemma to the process

one gets

Since

the third term is and can be dropped. We also have that

Applying Itō's lemma once again to , it follows that

The first term contains, in parentheses, the above PDE 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 Itō integral, which has expectation zero, it follows that

The desired result is obtained by observing that

and finally

Remarks[edit]

  • The proof above is essentially that of [1] with modifications to account for .
  • The expectation formula above is also valid for N-dimensional Itô diffusions. The corresponding PDE for becomes (see H. Pham book below):
where,
i.e. γ = σσ′, where σ′ denotes the transpose matrix of σ).
  • This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods.
  • When originally published by Kac in 1949,[2] 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).

See also[edit]

References[edit]

  • Simon, Barry (1979). Functional Integration and Quantum Physics. Academic Press. 
  • Hall, B. C. (2013). Quantum Theory for Mathematicians. Springer. 
  • Pham, Huyên (2009). Continuous-time stochastic control and optimisation with financial applications. Springer-Verlag. 
  1. ^ http://www.math.nyu.edu/faculty/kohn/pde_finance.html
  2. ^ 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 Mark Kac: Probability, Number Theory, and Statistical Physics, Selected Papers, edited by K. Baclawski and M.D. Donsker, The MIT Press, Cambridge, Massachusetts, 1979, pp.268-280