Feynman-Kac formula

Let $X_{t}$ be the $n$ -dimensional Itō process satisfying thestochastic differentialequation

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-differentiablefunction 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 respectto 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.