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.

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.

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:

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

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 $\ uV(x)\geq 0$,

$E(e^{-u\int _{0}^{t}V(x(\tau ))\,d\tau })=\int _{-\infty }^{\infty }w(x,t)\,dx$

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

${\frac {\partial w}{\partial t}}={\frac {1}{2}}{\frac {\partial ^{2}w}{\partial x^{2}}}-uV(x)w.$

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

${\frac {\partial w}{\partial t}}={\frac {1}{2}}{\frac {\partial ^{2}w}{\partial x^{2}}}-uV(x)w$

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

References

• Simon, Barry (1979). Functional Integration and Quantum Physics. Academic Press.