Feynman-Kac formula
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The Feynman-Kac formula establishes a link between partial differential equations (PDEs) and stochastic processes. It offers yet another method of solving certain PDEs: by simulating random paths of a stochastic process.
Suppose we are given the PDE
- <math>u_{t} + \mu(x,t) u_{x} + {1 \over 2} \sigma(x,t)^2 u_{xx} = 0 </math>
subject to the terminal condition
- <math>u(x,T)=\psi(x) </math>
where μ, σ2, ψ are known functions and u is the unknown. Then FK tells us that the solution can be written as an expectation:
- <math>u(x,t) = E[ \psi(X_T) | X(t)=x ] </math>
where X is an Itō process driven by the equation
- <math> dX = \mu(X,t)\,dt + \sigma(X,t)\,d W,</math>
where W is a Brownian motion and where the initial condition for X is X(t) = x. This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods
The FK formula can be proven by using Itō's lemma.
When originally published by Kac in 1949, the Feynman-Kac formula was originally presented as a formula for determining the distribution of certain Weiner 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( e^{- u \int_0^t V(x(\tau)) d\tau} )
= \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> p(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>. For more details, see "Functional Integration and Quantum Physics" by Barry Simon.