Annals of Probability

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

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
Revised: August 2019
First available in Project Euclid: 17 June 2020

https://projecteuclid.org/euclid.aop/1592359237

Digital Object Identifier
doi:10.1214/19-AOP1403

Mathematical Reviews number (MathSciNet)
MR4112723

Zentralblatt MATH identifier
07226369

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

References

• [1] Acemoglu, D., Ozdaglar, A. and Tahbaz-Salehi, A. (2014). Systemic Risk in Endogenous Financial Networks. Technical report, SSRN.
• [2] Ahuja, R. K., Magnanti, T. L. and Orlin, J. B. (1993). Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs, NJ.
• [3] Aliprantis, C. D. and Border, K. C. (2006). Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd ed. Springer, Berlin.
• [4] Atkeson, A. G., Eisfeldt, A. L. and Weill, P.-O. (2015). Entry and exit in OTC derivatives markets. Econometrica 83 2231–2292.
• [5] Aumann, R. J. (1964). Markets with a continuum of traders. Econometrica 32 39–50.
• [6] Babus, A. and Hu, T.-W. (2017). Endogenous intermediation in over-the-counter markets. J. Financ. Econ. 125 200–215.
• [7] Baccelli, F. L., Cohen, G., Olsder, G. J. and Quadrat, J.-P. (1992). Synchronization and Linearity: An Algebra for Discrete Event Systems. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, Chichester.
• [8] Baladron, J., Fasoli, D., Faugeras, O. and Touboul, J. (2012). Mean-field description and propagation of chaos in networks of Hodgkin–Huxley and FitzHugh–Nagumo neurons. J. Math. Neurosci. 2 Art. 10, 50.
• [9] Bhamidi, S., Budhiraja, A. and Wu, R. (2019). Weakly interacting particle systems on inhomogeneous random graphs. Stochastic Process. Appl. 129 2174–2206.
• [10] Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31 3–122.
• [11] Budhiraja, A. and Wu, R. (2016). Some fluctuation results for weakly interacting multi-type particle systems. Stochastic Process. Appl. 126 2253–2296.
• [12] Cáceres, M. J., Carrillo, J. A. and Perthame, B. (2011). Analysis of nonlinear noisy integrate & fire neuron models: Blow-up and steady states. J. Math. Neurosci. 1 Art. 7, 33.
• [13] Cáceres, M. J. and Perthame, B. (2014). Beyond blow-up in excitatory integrate and fire neuronal networks: Refractory period and spontaneous activity. J. Theoret. Biol. 350 81–89.
• [14] Capponi, A., Sun, X. and Yao, D. (2017). A Dynamic Network Model of Interbank Lending—Systemic Risk and Liquidity Provisioning. Technical report, SSRN.
• [15] Cardaliaguet, P. (2010). Notes on mean field games (from P.-L. Lions’ Lectures at Collège de France). Technical report. Available at https://www.ceremade.dauphine.fr/~cardalia/MFG20130420.pdf.
• [16] Carmona, R. and Delarue, F. (2018). Probabilistic Theory of Mean Field Games with Applications. II: Mean Field Games with Common Noise and Master Equations. Probability Theory and Stochastic Modelling 84. Springer, Cham.
• [17] Delarue, F. (2017). Mean field games: A toy model on an Erdös–Renyi graph. In Journées MAS 2016 de la SMAI—Phénomènes Complexes et Hétérogènes. ESAIM Proc. Surveys 60 1–26. EDP Sci., Les Ulis.
• [18] Delarue, F., Inglis, J., Rubenthaler, S. and Tanré, E. (2015). Global solvability of a networked integrate-and-fire model of McKean–Vlasov type. Ann. Appl. Probab. 25 2096–2133.
• [19] Delarue, F., Inglis, J., Rubenthaler, S. and Tanré, E. (2015). Particle systems with a singular mean-field self-excitation. Application to neuronal networks. Stochastic Process. Appl. 125 2451–2492.
• [20] Delarue, F., Nadtochiy, S. and Shkolnikov, M. (2019). Global solution to super-cooled Stefan problem with blow-ups: Regularity and uniqueness. Available at arXiv:1902.05174.
• [21] Dembo, A. and Tsai, L.-C. (2019). Criticality of a randomly-driven front. Arch. Ration. Mech. Anal. 233 643–699.
• [22] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38. Springer, Berlin.
• [23] Dudley, R. M. (2014). Uniform Central Limit Theorems, 2nd ed. Cambridge Studies in Advanced Mathematics 142. Cambridge Univ. Press, New York.
• [24] Elliott, M., Hazell, J. and Elliott, M. (2015). Endogenous Financial Networks: Efficient Modularity and Why Shareholders Prevent It. Technical report, SSRN.
• [25] Erol, S. and Vohra, R. (2017). Network Formation and Systemic Risk. Technical report, SSRN.
• [26] Farboodi, M. (2015). Intermediation and Voluntary Exposure to Counterparty Risk. Technical report, SSRN.
• [27] Fasano, A. and Primicerio, M. (1980/81). New results on some classical parabolic free-boundary problems. Quart. Appl. Math. 38 439–460.
• [28] Fasano, A. and Primicerio, M. (1983). A critical case for the solvability of Stefan-like problems. Math. Methods Appl. Sci. 5 84–96.
• [29] Fasano, A., Primicerio, M., Howison, S. D. and Ockendon, J. R. (1989). On the singularities of one-dimensional Stefan problems with supercooling. In Mathematical Models for Phase Change Problems (Óbidos, 1988). Internat. Ser. Numer. Math. 88 215–226. Birkhäuser, Basel.
• [30] Fasano, A., Primicerio, M., Howison, S. D. and Ockendon, J. R. (1990). Some remarks on the regularization of supercooled one-phase Stefan problems in one dimension. Quart. Appl. Math. 48 153–168.
• [31] Hajek, B. (1985). Mean stochastic comparison of diffusions. Z. Wahrsch. Verw. Gebiete 68 315–329.
• [32] Hambly, B., Ledger, S. and Søjmark, A. (2019). A McKean–Vlasov equation with positive feedback and blow-ups. Ann. Appl. Probab. 29 2338–2373.
• [33] Hambly, B. and Søjmark, A. (2019). An SPDE model for systemic risk with endogenous contagion. Finance Stoch. 23 535–594.
• [34] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
• [35] Knaster, B., Kuratowski, C. and Mazurkiewicz, S. (1929). Ein beweis des fixpunktsatzes für $n$-dimensionale simplexe. Fund. Math. 14 132–137.
• [36] Lasry, J.-M. and Lions, P.-L. (2007). Mean field games. Jpn. J. Math. 2 229–260.
• [37] McNamara, J. M. (1985). A regularity condition on the transition probability measure of a diffusion process. Stochastics 15 161–182.
• [38] Nadtochiy, S. and Shkolnikov, M. (2019). Particle systems with singular interaction through hitting times: Application in systemic risk modeling. Ann. Appl. Probab. 29 89–129.
• [39] Nagasawa, M. and Tanaka, H. (1987). Diffusion with interactions and collisions between coloured particles and the propagation of chaos. Probab. Theory Related Fields 74 161–198.
• [40] Neklyudov, A. and Sambalaibat, B. (2017). Endogenous Specialization and Dealer Networks. Technical report, SSRN.
• [41] Park, S. and Tan, D. H. (2000). Remarks on the Schauder–Tychonoff fixed point theorem. Vietnam J. Math. 28 127–132.
• [42] Skorohod, A. V. (1956). Limit theorems for stochastic processes. Teor. Veroyatn. Primen. 1 289–319.
• [43] Tychonoff, A. (1935). Ein Fixpunktsatz. Math. Ann. 111 767–776.
• [44] Wang, C. (2016). Core-Periphery Trading Networks. Technical report, SSRN.
• [45] Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer Series in Operations Research. Springer, New York.