Feynman-Kac formula

Let $X_{t}$ be the $n$ -dimensional Itō process satisfying the stochastic differential equation

$dX_{t}=\mu(X_{t})\,dt+\sigma(X_{t})\,dW_{t}$

and let $A$ be its infinitesimal generator.

Further suppose that $q$ is a lower-bounded continuous function on $\mathbb{R}^{n}$ , and $f$ is a twice-differentiable function on $\mathbb{R}^{n}$ with compact support. Then

$u(t,x)=\mathbb{E}\bigl{[}e^{-\int_{0}^{t}q(X_{s})\,ds}\,f(X_{t})\mid X_{0}=x% \bigr{]}\,,\quad t\geq 0\,,x\in\mathbb{R}^{n}$

is a solution to the partial differential equation

$\frac{\partial u}{\partial t}=Au(x)-uq(x)$

with initial condition $u(0,x)=f(x)$ .

(The expectation for $u$ is to be taken with respect to the probability measure under which $W_{t}$ is a Brownian motion.)

References

1 Bernt Øksendal. , An Introduction with Applications. 5th ed., Springer 1998.
2 Hui-Hsiung Kuo. Introduction to Stochastic Integration. Springer 2006.

Title	Feynman-Kac formula
Canonical name	FeynmanKacFormula
Date of creation	2013-03-22 17:16:11
Last modified on	2013-03-22 17:16:11
Owner	stevecheng (10074)
Last modified by	stevecheng (10074)
Numerical id	6
Author	stevecheng (10074)
Entry type	Theorem
Classification	msc 35K15
Classification	msc 60H30
Classification	msc 60H10
Related topic	RichardFeynman