Open Access
May 2020 Mean field systems on networks, with singular interaction through hitting times
Sergey Nadtochiy, Mykhaylo Shkolnikov
Ann. Probab. 48(3): 1520-1556 (May 2020). DOI: 10.1214/19-AOP1403


Building on the line of work (Ann. Appl. Probab. 25 (2015) 2096–2133; Stochastic Process. Appl. 125 (2015) 2451–2492; Ann. Appl. Probab. 29 (2019) 89–129; Arch. Ration. Mech. Anal. 233 (2019) 643–699; Ann. Appl. Probab. 29 (2019) 2338–2373; Finance Stoch. 23 (2019) 535–594), we continue the study of particle systems with singular interaction through hitting times. In contrast to the previous research, we (i) consider very general driving processes and interaction functions, (ii) allow for inhomogeneous connection structures and (iii) analyze a game in which the particles determine their connections strategically. Hereby, we uncover two completely new phenomena. First, we characterize the “times of fragility” of such systems (e.g., the times when a macroscopic part of the population defaults or gets infected simultaneously, or when the neuron cells “synchronize”) explicitly in terms of the dynamics of the driving processes, the current distribution of the particles’ values and the topology of the underlying network (represented by its Perron–Frobenius eigenvalue). Second, we use such systems to describe a dynamic credit-network game and show that, in equilibrium, the system regularizes, that is, the times of fragility never occur, as the particles avoid them by adjusting their connections strategically. Two auxiliary mathematical results, useful in their own right, are uncovered during our investigation: a generalization of Schauder’s fixed-point theorem for the Skorokhod space with the M1 topology, and the application of max-plus algebra to the equilibrium version of the network flow problem.


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Sergey Nadtochiy. Mykhaylo Shkolnikov. "Mean field systems on networks, with singular interaction through hitting times." Ann. Probab. 48 (3) 1520 - 1556, May 2020.


Received: 1 November 2018; Revised: 1 August 2019; Published: May 2020
First available in Project Euclid: 17 June 2020

zbMATH: 07226369
MathSciNet: MR4112723
Digital Object Identifier: 10.1214/19-AOP1403

Primary: 05C57 , 82C22
Secondary: 54H25

Keywords: cascades , credit network game , directed weighted graphs , dynamic games , M1 topology , max-plus algebra , mean field games on graphs , Nash equilibrium , network flow problem , Particle systems , Perron–Frobenius eigenvalue , regularization through a game , Schauder’s fixed-point theorem , self-excitation , singular interaction through hitting times , systemic risk , times of fragility

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 3 • May 2020
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