# Feynman-Kac formula

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

$$d{X}_{t}=\mu ({X}_{t})dt+\sigma ({X}_{t})d{W}_{t}$$ |

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

$$u(t,x)=\mathbb{E}[{e}^{-{\int}_{0}^{t}q({X}_{s})\mathit{d}s}f({X}_{t})\mid {X}_{0}=x],t\ge 0,x\in {\mathbb{R}}^{n}$$ |

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 |