McKean Feynman-Kac probabilistic representations of non-linear partial differential equations

TL;DR

This paper reviews the use of McKean Feynman-Kac equations to represent complex non-linear PDEs, highlighting their applications in modeling, numerical approximation, and stochastic control.

Contribution

It introduces the concept of MFKEs as a generalization of MSDEs for representing non-conservative non-linear PDEs and discusses their relevance in various applications.

Findings

MFKEs extend MSDEs to non-conservative PDEs

MFKEs are useful in stochastic control modeling

Representation of backward Fokker-Planck equations

Abstract

This paper presents a partial state of the art about the topic of representation of generalized Fokker-Planck Partial Differential Equations (PDEs) by solutions of McKean Feynman-Kac Equations (MFKEs) that generalize the notion of McKean Stochastic Differential Equations (MSDEs). While MSDEs can be related to non-linear Fokker-Planck PDEs, MFKEs can be related to non-conservative non-linear PDEs. Motivations come from modeling issues but also from numerical approximation issues in computing the solution of a PDE, arising for instance in the context of stochastic control. MFKEs also appear naturally in representing final value problems related to backward Fokker-Planck equations.