Feynman-Kac formula

The Feynman-Kac formula, named after Richard Feynman and Mark Kac, establishes a link between partial differential equations (PDEs) and stochastic processes. It offers a method of solving certain PDEs by simulating random paths of a stochastic process. Conversely, stochastic PDEs can be solved by deterministic methods.

Suppose we are given the PDE

:$frac\{partial\; f\}\{partial\; t\}\; +\; mu(x,t)\; frac\{partial\; f\}\{partial\; x\}\; +\; frac\{1\}\{2\}\; sigma^2(x,t)\; frac\{partial^2\; f\}\{partial\; x^2\}\; =\; 0$

subject to the terminal condition

:$f(x,T)=psi(x)$

where $mu,\; sigma,\; psi$ are known functions, $T$ is a parameter and $f$ is the unknown. This is known as the (one-dimensional) Kolmogorov backward equation. Then the Feynman-Kac formula tells us that the solution can be written as an expectation:

:$f(x,t)\; =\; E\; [\; psi(X\_T)\; |\; X\_t=x\; ]$

where $X$ is an Itō process driven by the equation

:$dX\; =\; mu(X,t),dt\; +\; sigma(X,t),dW,$

where $W(t)$ is a Wiener process (also called Brownian motion) and the initial condition for $X(t)$ is $X(0)\; =\; x$. This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods.

Proof

Applying Itō's lemma to the unknown function $f$ one gets

:$df=left(mu(x,t)frac\{partial\; f\}\{partial\; x\}+frac\{partial\; f\}\{partial\; t\}+frac\{1\}\{2\}sigma^2(x,t)frac\{partial^2\; f\}\{partial\; x^2\}\; ight),dt+sigma(x,t)frac\{partial\; f\}\{partial\; x\},dW.$

The first term in parentheses is the above PDE and is zero by hypothesis. Integrating both sides one gets

:$int\_t^T\; df=f(X\_T,T)-f(x,t)=int\_t^Tsigma(x,t)frac\{partial\; f\}\{partial\; x\},dW.$

Reorganising and taking the expectation of both sides:

:$f(x,t)=\; extrm\{E\}left\; [f(X\_T,T)\; ight]\; -\; extrm\{E\}left\; [int\_t^Tsigma(x,t)frac\{partial\; f\}\{partial\; x\},dW\; ight]\; .$

Since the expectation of an Itō integral with respect to a Wiener process $W$ is zero, one gets the desired result:

:$f(x,t)=\; extrm\{E\}left\; [f(X\_T,T)\; ight]\; =\; extrm\{E\}left\; [psi(X\_T)\; ight]\; =\; extrm\{E\}left\; [psi(X\_T)|X\_t=x\; ight]\; .$

Remarks

When originally published by Kac in 1949 [cite journal|last=Kac|first=Mark|title=On Distributions of Certain Wiener Functionals|journal=Transactions of the American Mathematical Society|authorlink=Mark Kac|volume=65|issue=1|pages=1–13|url=http://www.jstor.org/stable/1990512|date=1949|accessdate=2008-05-30|doi=10.2307/1990512] , 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

:$e^\{-int\_0^t\; V(x(\; au)),\; d\; au\}$

in the case where $x(\; au)$ 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 adiffusion equation. Specifically, under the conditions that $u\; V(x)\; geq\; 0$,

:$Eleft(\; e^\{-\; u\; int\_0^t\; V(x(\; au)),\; d\; au\}\; ight)\; =\; int\_\{-infty\}^\{infty\}\; w(x,t),\; dx$

where $w(x,0)\; =\; delta(x)$ and

:$frac\{partial\; w\}\{partial\; t\}\; =\; frac\{1\}\{2\}\; frac\{partial^2\; w\}\{partial\; x^2\}\; -\; u\; V(x)\; w.$

The Feynman-Kac formula can also be interpreted as a method for evaluating functional integrals of a certain form. If

:$I\; =\; int\; f(x(0))\; e^\{-uint\_0^t\; V(x(t)),\; dt\}\; g(x(t)),\; Dx$

where the integral is taken over all random walks, then

:$I\; =\; int\; w(x,t)\; g(x),\; dx$

where $w(x,t)$ is a solution to the parabolic partial differential equation

:$frac\{partial\; w\}\{partial\; t\}\; =\; frac\{1\}\{2\}\; frac\{partial^2\; w\}\{partial\; x^2\}\; -\; u\; V(x)\; w$

with initial condition $w(x,0)\; =\; f(x)$.

See also

* Itō's lemma
* Kunita-Watanabe theorem
* Girsanov theorem
* Kolmogorov forward equation (also known as Fokker-Planck equation)

References

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