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February 2019 Particle systems with singular interaction through hitting times: Application in systemic risk modeling
Sergey Nadtochiy, Mykhaylo Shkolnikov
Ann. Appl. Probab. 29(1): 89-129 (February 2019). DOI: 10.1214/18-AAP1403

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 noncore 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 noncore 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 [Ann. Appl. Probab. 25 (2015) 2096–2133] and [Stochastic Process. Appl. 125 (2015) 2451–2492], 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.

Citation

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Sergey Nadtochiy. Mykhaylo Shkolnikov. "Particle systems with singular interaction through hitting times: Application in systemic risk modeling." Ann. Appl. Probab. 29 (1) 89 - 129, February 2019. https://doi.org/10.1214/18-AAP1403

Information

Received: 1 May 2017; Revised: 1 November 2017; Published: February 2019
First available in Project Euclid: 5 December 2018

zbMATH: 07039122
MathSciNet: MR3910001
Digital Object Identifier: 10.1214/18-AAP1403

Subjects:
Primary: 35B65, 35K20, 82C22, 91G80

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 1 • February 2019
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