Path-dependent McKean PDEs with reaction: a discussion on probabilistic interpretations and particle approximations

TL;DR

This paper compares two probabilistic methods for linking path-dependent McKean PDEs with reaction terms to stochastic differential equations, exploring their interpretations, models, and particle approximation techniques.

Contribution

It introduces and analyzes two distinct probabilistic frameworks for McKean PDEs with reactions, highlighting their differences and implications for particle system approximations.

Findings

Two interpretations of sub-probability measures are compared.

Different microscopic SDE models are derived from each interpretation.

Empirical densities from particle systems serve as kernel estimators for PDE solutions.

Abstract

In this paper, we discuss and compare two probabilistic approaches for associating a stochastic differential equation with a McKean-type partial differential equation featuring a reaction term and path-dependent coefficients. The non-conservative nature of the macroscopic dynamics leads to two possible interpretations of the sub-probability measure and of the associated SDE equation at the microscale: on the one hand, as a measure-valued solution of a Feynman-Kac-type equation; on the other hand, as the sub-probability associated with an SDE defined up to a survival time with a reaction-dependent rate. These different interpretations give rise to two different microscopic stochastic models and therefore to two different techniques of probabilistic analysis. Finally, by considering the interacting particle systems associated with both models, we discuss how their empirical densities provide two different kernel estimators for the PDE solution.