Annals of Probability

Mean field systems on networks, with singular interaction through hitting times

Sergey Nadtochiy and Mykhaylo Shkolnikov

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Abstract

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.

Article information

Source
Ann. Probab., Volume 48, Number 3 (2020), 1520-1556.

Dates
Received: November 2018
Revised: August 2019
First available in Project Euclid: 17 June 2020

Permanent link to this document
https://projecteuclid.org/euclid.aop/1592359237

Digital Object Identifier
doi:10.1214/19-AOP1403

Mathematical Reviews number (MathSciNet)
MR4112723

Zentralblatt MATH identifier
07226369

Subjects
Primary: 82C22: Interacting particle systems [See also 60K35] 05C57: Games on graphs [See also 91A43, 91A46]
Secondary: 54H25: Fixed-point and coincidence theorems [See also 47H10, 55M20]

Keywords
Cascades credit network game directed weighted graphs dynamic games max-plus algebra mean field games on graphs M1 topology 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

Citation

Nadtochiy, Sergey; Shkolnikov, Mykhaylo. Mean field systems on networks, with singular interaction through hitting times. Ann. Probab. 48 (2020), no. 3, 1520--1556. doi:10.1214/19-AOP1403. https://projecteuclid.org/euclid.aop/1592359237


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