# 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 FeynmanKacFormula 2013-03-22 17:16:11 2013-03-22 17:16:11 stevecheng (10074) stevecheng (10074) 6 stevecheng (10074) Theorem msc 35K15 msc 60H30 msc 60H10 RichardFeynman