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Mean field systems on networks, with singular interaction through hitting times. (English) Zbl 1448.82032
The article belongs to a group of works dealing with the investigation of the mean field particle sytems with singular interaction through hitting times. The second section contains a detailed description of the proposed particle systems and its dynamics. The system of particles is characterized by their “level of healthiness” \(Y^x\), taking values in \([0,\infty)\) and by their “type” \(x\in\mathcal{X}\), \(\mathcal{X}\) an abstract finite set. These “level of healthiness “ are defined as a family of sochastic processes with a dynamics of the form \({Y_t}^x = {Z_t}^x + C(x) \int_{\mathcal{X}}g({\mathbb{P}}({\tau}^{x'}>t)){\kappa}(x,dx')\). The interaction function \(g\) is assumed continuous and nondecreasing on the interval \((0,1]\).
As practical interpretation of the system, the particles may represent cells, individuals or organisations with exposures to each other, for instance a neuron in the human brain is physically connected to neighbouring neurons or a computer can be part of a network. Under a special assumption the existence of a solution to the considered model is proved. The uniqueness of the solution is not part of the investigation. In the third section the times of fragility of the considered particle systems are studied. Necessary and sufficient conditions for fragility are proved. The forth section is devoted to the study of controlled mean field dynamics and equilibrium. In the considered game, every particle is allowed to control the strength of its connections dynamically, however by optimizing its objective. The optimization problem and the notion of equilibrium are introduced, one proves the existence of an equilibrium and one developes a technique to its construction. In the second section a generalization of Schauder’s fixed-point theorem for mappings defined on the Skorohod space with the M1 tolopogy is proved. Proofs and complementary results are contained in Appendix A and Appendix B.
82C22 Interacting particle systems in time-dependent statistical mechanics
05C57 Games on graphs (graph-theoretic aspects)
54H25 Fixed-point and coincidence theorems (topological aspects)
92C20 Neural biology
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[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. · Zbl 1201.90001
[3] Aliprantis, C. D. and Border, K. C. (2006). Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd ed. Springer, Berlin. · Zbl 1156.46001
[4] Atkeson, A. G., Eisfeldt, A. L. and Weill, P.-O. (2015). Entry and exit in OTC derivatives markets. Econometrica 83 2231-2292. · Zbl 1419.91603
[5] Aumann, R. J. (1964). Markets with a continuum of traders. Econometrica 32 39-50. · Zbl 0137.39003
[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. · Zbl 0824.93003
[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. · Zbl 1322.60205
[9] Bhamidi, S., Budhiraja, A. and Wu, R. (2019). Weakly interacting particle systems on inhomogeneous random graphs. Stochastic Process. Appl. 129 2174-2206. · Zbl 07074604
[10] Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31 3-122. · Zbl 1123.05083
[11] Budhiraja, A. and Wu, R. (2016). Some fluctuation results for weakly interacting multi-type particle systems. Stochastic Process. Appl. 126 2253-2296. · Zbl 1338.60228
[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. · Zbl 1259.35198
[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. · Zbl 1412.92001
[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. · Zbl 1422.91015
[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. · Zbl 1407.91054
[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. · Zbl 1322.60085
[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. · Zbl 1328.60134
[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. · Zbl 07061534
[22] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38. Springer, Berlin. · Zbl 1177.60035
[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. · Zbl 0468.35081
[28] Fasano, A. and Primicerio, M. (1983). A critical case for the solvability of Stefan-like problems. Math. Methods Appl. Sci. 5 84-96. · Zbl 0526.35078
[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. · Zbl 0699.35266
[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. · Zbl 0703.35178
[31] Hajek, B. (1985). Mean stochastic comparison of diffusions. Z. Wahrsch. Verw. Gebiete 68 315-329. · Zbl 0537.60050
[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. · Zbl 1423.35224
[33] Hambly, B. and Søjmark, A. (2019). An SPDE model for systemic risk with endogenous contagion. Finance Stoch. 23 535-594. · Zbl 07074031
[34] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York. · Zbl 0734.60060
[35] Knaster, B., Kuratowski, C. and Mazurkiewicz, S. (1929). Ein beweis des fixpunktsatzes für \(n\)-dimensionale simplexe. Fund. Math. 14 132-137. · JFM 55.0972.01
[36] Lasry, J.-M. and Lions, P.-L. (2007). Mean field games. Jpn. J. Math. 2 229-260. · Zbl 1156.91321
[37] McNamara, J. M. (1985). A regularity condition on the transition probability measure of a diffusion process. Stochastics 15 161-182. · Zbl 0581.60061
[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. · Zbl 1417.35204
[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. · Zbl 0587.60095
[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. · Zbl 0970.47045
[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. · Zbl 0012.30803
[44] Wang, C. (2016). Core-Periphery Trading Networks. Technical report, SSRN.
[45] Whitt, W. · Zbl 0993.60001
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