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
- <math>\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 </math>
subject to the terminal condition
- <math>\ f(x,T)=\psi(x) </math>
where <math>\mu,\ \sigma,\ \psi</math> are known functions, <math>\ T</math> is a parameter and <math>\ f</math> 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:
- <math>\ f(x,t) = E[ \psi(X_T) | X_t=x ] </math>
where <math>\ X</math> is an Itō process driven by the equation
- <math>dX = \mu(X,t)\,dt + \sigma(X,t)\,dW,</math>
where <math>\ W(t)</math> is a Wiener process (also called Brownian motion) and the initial condition for <math>\ X(t)</math> is <math>\ X(0) = x</math>. This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods.
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Proof
Applying Itō's lemma to the unknown function <math>\ f</math> one gets
- <math>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.</math>
The first term in parentheses is the above PDE and is zero by hypothesis. Integrating both sides one gets
- <math>\int_t^T df=f(X_T,T)-f(x,t)=\int_t^T\sigma(x,t)\frac{\partial f}{\partial x}\,dW.</math>
Reorganising and taking the expectation of both sides:
- <math>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]</math>
Since the expectation of an Itō integral with respect to a Wiener process <math>\ W</math> is zero, one gets the desired result:
- <math>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]</math>
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
- <math> e^{-\int_0^t V(x(\tau))\, d\tau} </math>
in the case where <math>\ x(\tau)</math> is some realization of a diffusion process starting at <math>\ x(0) = 0</math>. 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 <math>\ u V(x) \geq 0</math>,
- <math> E\left( e^{- u \int_0^t V(x(\tau))\, d\tau} \right) = \int_{-\infty}^{\infty} w(x,t)\, dx </math>
where <math>\ w(x,0) = \delta(x)</math> and
- <math>
\frac{\partial w}{\partial t} = \frac{1}{2} \frac{\partial^2 w}{\partial x^2} - u V(x) w. </math> The Feynman-Kac formula can also be interpreted as a method for evaluating functional integrals of a certain form. If
- <math> I = \int f(x(0)) e^{-u\int_0^t V(x(t))\, dt} g(x(t))\, Dx </math>
where the integral is taken over all random walks, then
- <math> I = \int w(x,t) g(x)\, dx </math>
where <math>\ w(x,t)</math> is a solution to the parabolic partial differential equation
- <math> \frac{\partial w}{\partial t} = \frac{1}{2} \frac{\partial^2 w}{\partial x^2} - u V(x) w </math>
with initial condition <math>\ w(x,0) = f(x)</math>.
See also
- Richard Feynman
- Mark Kac
- Itō's lemma
- Kunita-Watanabe theorem
- Girsanov theorem
- Kolmogorov forward equation (also known as Fokker-Planck equation)
References
- Simon, Barry (1979). Functional Integration and Quantum Physics. Academic Press.
