Directed Chain Stochastic Differential Equations

TL;DR

This paper introduces a novel particle system model with chain-like and mean-field interactions, analyzing its properties and the conditions under which mean-field effects can be detected from limited observations.

Contribution

It formulates a new class of stochastic differential equations combining local chain and mean-field interactions, and explores their properties and observational detectability.

Findings

The particle system can be approximated by a finite particle limit.

Propagation of chaos may not hold due to local interactions.

Detection of mean-field interaction from a single component is discussed.

Abstract

We propose a particle system of diffusion processes coupled through a chain-like network structure described by an infinite-dimensional, nonlinear stochastic differential equation of McKean-Vlasov type. It has both (i) a local chain interaction and (ii) a mean-field interaction. It can be approximated by a limit of finite particle systems, as the number of particles goes to infinity. Due to the local chain interaction, propagation of chaos does not necessarily hold. Furthermore, we exhibit a dichotomy of presence or absence of mean-field interaction, and we discuss the problem of detecting its presence from the observation of a single component process.