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


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

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


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

Reorganising and taking the expectation of both sides:


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

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


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

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 $$\ u V(x) \geq 0$$,


 * $$ E\left( e^{- u \int_0^t V(x(\tau))\, d\tau} \right) = \int_{-\infty}^{\infty} w(x,t)\, dx $$

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


 * $$ I = \int f(x(0)) e^{-u\int_0^t V(x(t))\, dt} g(x(t))\, Dx $$

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 $$\ w(x,0) = f(x)$$.