Feynman-Kac formula


Let Xt be the n-dimensional Itō process satisfying the stochastic differential equation

dXt=μ(Xt)dt+σ(Xt)dWt

and let A be its infinitesimal generator.

Further suppose that q is a lower-bounded continuous functionMathworldPlanetmath on n, and f is a twice-differentiable function on n with compact support. Then

u(t,x)=𝔼[e-0tq(Xs)𝑑sf(Xt)X0=x],t0,xn

is a solution to the partial differential equationMathworldPlanetmath

ut=Au(x)-uq(x)

with initial conditionMathworldPlanetmath u(0,x)=f(x).

(The expectation for u is to be taken with respect to the probability measureMathworldPlanetmath under which Wt is a Brownian motionMathworldPlanetmath.)

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