Particle Systems with Local Interactions via Hitting Times and Cascades on Graphs

TL;DR

This paper develops a rigorous framework for modeling systemic risk in financial networks using particle systems with local interactions via hitting times, capturing default cascades and systemic events in sparse graphs.

Contribution

It extends convergence theory to systems with singular interactions, connecting fragility to percolation thresholds and analyzing default times in tree-like networks.

Findings

Established conditions for well-posedness of the model

Proved convergence of empirical distributions

Characterized default time distribution in tree networks

Abstract

We introduce a family of particle systems on sparse graphs where local interactions occur via hitting times, providing a dynamic and tractable model for default cascades in large sparsely-connected financial networks. Building on the framework of Lacker, Ramanan and Wu (2023), we extend convergence theory to systems with singular interactions, capturing the abrupt and discontinuous nature of systemic events. We establish conditions for well-posedness through a minimality principle and connect fragility to dynamic percolation thresholds. Our analysis demonstrates continuity of the joint law of defaults with respect to local graph convergence, establishes convergence of empirical distributions, and characterizes the default time distribution in tree-like networks. This framework offers a rigorous and flexible foundation for modeling systemic risk in evolving financial systems, featuring continuous-time dynamics, heterogeneous and local interactions, and instantaneous default cascades.