Feynman-Kac formula

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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.

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) Fokker-Planck equation or Kolmogorov forward 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 $\ X(t) = x$ . This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods.

[edit] Proof

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

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

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

$\int_t^T df=f(x,T)-f(x,t)=\int_t^T\sigma(x,t)\frac{\partial f}{\partial x}\,dW.$

Reorganising and taking the expectation of both sides:

$f(x,t)=\textrm{E}\left[f(x,T)\right]-\textrm{E}\left[\int_t^T\sigma(x,t)\frac{\partial f}{\partial x}\,dW\right]$

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

$f(x,t)=\textrm{E}\left[f(x,T)\right]=\textrm{E}\left[\psi(x)\right]=\textrm{E}\left[\psi(X_T)|X_t=x\right]$

[edit] Remarks

When originally published by Kac in 1949, 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

$e^{-\int_0^t V(x(\tau))\, d\tau}$

in the case where $\ x(\tau)$ 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 $\ u V(x) \geq 0$ ,