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 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

defined for all and , 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 Itô 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]

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 u(x, t) 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 t to T, one concludes that

Upon taking expectations, conditioned on Xt = x, and observing that the right side is an Itô integral, which has expectation zero[citation needed], it follows that

The desired result is obtained by observing that

and finally

Remarks[edit]

  • The proof above that a solution must have the given form is essentially that of [1] with modifications to account for .
  • The expectation formula above is also valid for N-dimensional Itô diffusions. The corresponding partial differential equation for becomes:[2]
where,
i.e. , where denotes the transpose of .
  • This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods.
  • When originally published by Kac in 1949,[3] 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).

Applications[edit]

In quantitative finance, the Feynman–Kac formula is used to efficiently calculate solutions to the Black–Scholes equation to price options on stocks.[4]

See also[edit]

References[edit]

  1. ^ http://www.math.nyu.edu/faculty/kohn/pde_finance.html
  2. ^ See Pham, Huyên (2009). Continuous-time stochastic control and optimisation with financial applications. Springer-Verlag. ISBN 978-3-642-10044-4. 
  3. ^ 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. 
  4. ^ 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. 

Further reading[edit]

  • Simon, Barry (1979). Functional Integration and Quantum Physics. Academic Press. 
  • Hall, B. C. (2013). Quantum Theory for Mathematicians. Springer.