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

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

subject to the terminal condition

u (x, T) = ψ(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 [ψ(X T) | X = X 0]

where X is an Itô process driven by the equation

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

This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods

Feynman-Kac formula

See also