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

$u_{t}+\mu (x,t)u_{x}+{1 \over 2}\sigma (x,t)^{2}u_{xx}=0$

subject to the terminal condition

$u(x,T)=\psi (x)$

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:

$u=E[\psi (X_{T})|X=X_{0}]$

where X is an Itō process driven by the equation

$dX=\mu (X,t)\,dt+\sigma (X,t)\,dW,$

where W is a Brownian motion. This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods

The FK formula can be proven by using Ito's lemma.

See also

• Itō's lemma
• Richard Feynman
• Mark Kac