A Feynman-Kac Type Theorem for ODEs: Solutions of Second Order ODEs as Modes of Diffusions

In this article, we prove a Feynman-Kac type result for a broad class of second order ordinary differential equations. The classical Feynman-Kac theorem says that the solution to a broad class of second order parabolic equations is the mean of a particular diffusion. In our situation, we show that t...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2021-06
Main Authors: Selk, Zachary, Honnappa, Harsha
Format: Article
Language:English
Subjects:
Differential equations
Monte Carlo simulation
Ordinary differential equations
Theorems
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:In this article, we prove a Feynman-Kac type result for a broad class of second order ordinary differential equations. The classical Feynman-Kac theorem says that the solution to a broad class of second order parabolic equations is the mean of a particular diffusion. In our situation, we show that the solution to a system of second order ordinary differential equations is the mode of a diffusion, defined through the Onsager-Machlup formalism. One potential utility of our result is to use Monte Carlo type methods to estimate the solutions of ordinary differential equations. We conclude with examples of our result illustrating its utility in numerically solving linear second order ODEs.
ISSN:2331-8422