# Particle systems with singular interaction through hitting times: application in systemic risk modeling

## TL;DR

This paper models systemic risk in banking systems using a particle system where defaults cause cascades, analyzing large-system limits and the impact of singular interactions on systemic events.

## Contribution

It introduces a novel particle system with singular interactions to model systemic risk, extending previous models and characterizing large-population behavior and default cascades.

## Key findings

- Discontinuities in loss processes represent systemic events.
- Large-population limit characterized with jump times and regularity.
- Singular interactions significantly influence cascade dynamics.

## Abstract

We propose an interacting particle system to model the evolution of a system of banks with mutual exposures. In this model, a bank defaults when its normalized asset value hits a lower threshold, and its default causes instantaneous losses to other banks, possibly triggering a cascade of defaults. The strength of this interaction is determined by the level of the so-called non-core exposure. We show that, when the size of the system becomes large, the cumulative loss process of a bank resulting from the defaults of other banks exhibits discontinuities. These discontinuities are naturally interpreted as systemic events, and we characterize them explicitly in terms of the level of non-core exposure and the fraction of banks that are "about to default". The main mathematical challenges of our work stem from the very singular nature of the interaction between the particles, which is inherited by the limiting system. A similar particle system is analyzed in [DIRT15a] and [DIRT15b], and we build on and extend their results. In particular, we characterize the large-population limit of the system and analyze the jump times, the regularity between jumps, and the local uniqueness of the limiting process.