Feynman-Kac formula
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 = E[ \psi(X_T) | X=X_0 ] <math>
where X is an Itô process driven by the equation
- <math> dX = \mu(X,t)\,dt + \sigma(X,t)\,dZ.<math>
This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods