Annals of Finance

, Volume 15, Issue 4, pp 489–538 | Cite as

Dynamic contagion in a banking system with births and defaults

  • Tomoyuki Ichiba
  • Michael Ludkovski
  • Andrey SarantsevEmail author
Research Article


We consider a dynamic model of interconnected banks. New banks can emerge, and existing banks can default, creating a birth-and-death setup. Microscopically, banks evolve as independent geometric Brownian motions. Systemic effects are captured through default contagion: as one bank defaults, reserves of other banks are reduced by a random proportion. After examining the long-term stability of this system, we investigate mean-field limits as the number of banks tends to infinity. Our main results concern the measure-valued scaling limit which is governed by a McKean–Vlasov jump-diffusion. The default impact creates a mean-field drift, while the births and defaults introduce jump terms tied to the current distribution of the process. Individual dynamics in the limit is described by the propagation of chaos phenomenon. In certain cases, we explicitly characterize the limiting average reserves.


Default contagion Mean field limit Interacting birth-and-death process McKean–Vlasov jump-diffusion Propagation of chaos Lyapunov function 

Mathematics Subject Classification

60J70 60J75 60K35 91B70 

JEL Classification

C60 E17 G10 O41 



Part of the research was supported by National Science Foundation under grants NSF DMS-1615229, NSF DMS-1521743, and NSF DMS-1409434. The authors are thankful to the referee and the editors for their comments and suggestions. Sarantsev benefited from the discussion with Clayton Barnes, Ricardo Fernholz, and Mykhaylo Shkolnikov.


  1. Andreis, L., Dai Pra, P., Fischer, M.: Mckean–Vlasov limit for interacting systems with simultaneous jumps. Stoch Anal Appl 36, 960–995 (2018)CrossRefGoogle Scholar
  2. Bass, Richard F.: Adding and subtracting jumps from Markov processes. Trans Am Math Soc 255, 363–376 (1979)CrossRefGoogle Scholar
  3. Bo, L., Capponi, A.: Bilateral credit valuation adjustment for large credit derivatives portfolios. Finance Stoch 12, 43–482 (2014)Google Scholar
  4. Bo, L., Capponi, A.: Systemic risk in interbanking networks. SIAM J Financ Math 6(1), 386–424 (2015)CrossRefGoogle Scholar
  5. Benazzoli, C., Campi, L., Di Persio, L.: Mean-field games with controlled jumps (2017a). arXiv:1703.01919
  6. Benazzoli, C., Campi, L., Di Persio, L. \(\varepsilon \)-nash equilibrium in stochastic differential games with mean-field interaction and controlled jumps (2017b). arXiv:1710.05734
  7. Campi, L., Fischer, M.: \(N\)-player games and mean-field games with absorption. Ann Appl Probab 28(4), 2188–2242 (2018)CrossRefGoogle Scholar
  8. Cvitanić, J., Ma, J., Zhang, J.: The law of large numbers for self-exciting correlated defaults. Stoch Proc Appl 122(8), 2781–2810 (2012)CrossRefGoogle Scholar
  9. Capponi, A., Sun, X., Yao. D.: A dynamic network model of interbank lending—systemic risk and liquidity provisioning (2019). PreprintGoogle Scholar
  10. Das, S.R., Duffie, D., Kapadia, N., Saita, L.: Common failings: how corporate defaults are correlated. J Finance 62, 93–117 (2007)CrossRefGoogle Scholar
  11. Delarue, F., Inglis, J., Rubenthaler, S., Tanré, E.: Global solvability of a networked integrate-and-fire model of McKean–Vlasov type. Ann Appl Probab 25(4), 2096–2133 (2015a)CrossRefGoogle Scholar
  12. Delarue, F., Inglis, J., Rubenthaler, S., Tanré, E.: Particle systems with a singular mean-field self-excitation. Application to neuronal networks. Stoch Proc Appl 125(6), 2451–2492 (2015b)CrossRefGoogle Scholar
  13. De Masi, A., Galves, A., Löcherbach, E., Presutti, E.: Hydrodynamic limit for interacting neurons. J Stat Phys 158(4), 866–902 (2015)CrossRefGoogle Scholar
  14. Down, D.G., Meyn, S.P., Tweedie, R.L.: Exponential and uniform ergodicity of Markov processes. Ann Probab 23(4), 1671–1691 (1995)CrossRefGoogle Scholar
  15. Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics. Wiley, Hoboken (1986)CrossRefGoogle Scholar
  16. Fouque, J.-P., Ichiba, T.: Stability in a model of interbank lending. SIAM J Financ Math 4(1), 784–803 (2013a)CrossRefGoogle Scholar
  17. Fouque, J.-P., Langsam, J.: Handbook of Systemic Risk. Cambridge University Press, Cambridge (2013)Google Scholar
  18. Fournier, N., Löcherbach, E.: On a toy model of interacting neurons. Ann Inst Henri Poincaré Probab Stat 52(4), 1844–1876 (2016)CrossRefGoogle Scholar
  19. Folland, G.B.: Real Analysis. Pure and Applied Mathematics, 2nd edn. Wiley, New York (1999)Google Scholar
  20. Funaki, T.: A certain class of diffusion processes associated with nonlinear parabolic equations. Probab Theory Relat Fields 67(3), 331–348 (1984)Google Scholar
  21. Graham, C.: McKean-Vlasov Itô-Skorohod equations, and nonlinear diffusions with discrete jump sets. Stoch Proc Appl 40(1), 69–82 (1992a)CrossRefGoogle Scholar
  22. Graham, C.: Nonlinear diffusion with jumps. Ann Inst Henri Poincaré Probab Stat 28(3), 393–402 (1992b)Google Scholar
  23. Giesecke, K., Spiliopoulos, K., Sowers, R.B.: Default clustering in large portfolios: typical events. Ann Appl Probab 23(1), 348–385 (2013)CrossRefGoogle Scholar
  24. Hambly, B. , Sojmark, A.: An SPDE model for systemic risk with endogenous contagion. Finance Stoch (2018a). arXiv:1801.10088. To appear
  25. Hambly, B., Ledger, S., Sojmark. A.: A McKean–Vlasov equation with positive feedback and blow-ups. Ann Appl Probab (2018). arXiv:1801.07703. To appear
  26. Ichiba, T., Sarantsev, A.: Convergence and stationary distributions for Walsh diffusions. Bernoulli (2018). arXiv:1706.07127. To appear
  27. Kaushansky, V., Reisinger, C.: Simulation of particle systems interacting through hitting times. arXiv:1805.11678 (2018)
  28. Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculu. Graduate Texts in Mathematics, vol. 113, 2nd edn. Springer, Berlin (1991)Google Scholar
  29. Karatzas, I., Sarantsev, A.: Diverse market models of competing Brownian particles with splits and mergers. Ann Appl Probab 26(3), 1329–1361 (2016)CrossRefGoogle Scholar
  30. Lipton, A., Kaushansky, V., Reisinger, C.: Semi-analytical solution of a mckean-vlasov equation with feedback through hitting a boundary (2018). arXiv:1808.05311
  31. Lund, R.B., Meyn, S.P., Tweedie, R.L.: Computable exponential convergence rates for stochastically ordered Markov processes. Ann Appl Probab 6(1), 218–237 (1996)CrossRefGoogle Scholar
  32. Mehri, S., Scheutzow, M., Stannat, W., Zangeneh, B.Z.: Propagation of chaos for stochastic spatially structured neuronal networks with fully path dependent delays and monotone coefficients driven by jump diffusion noise (2018). arXiv:1805.01654
  33. Meyn, S.P., Tweedie, R.L.: Stability of Markovian processes. II. Continuous-time processes and sampled chains. Adv Appl Probab 25(3), 487–517 (1993a)CrossRefGoogle Scholar
  34. Meyn, S.P., Tweedie, R.L.: Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes. Adv Appl Probab 25(3), 518–548 (1993b)CrossRefGoogle Scholar
  35. Nadtochiy, S., Shkolnikov, M.: Particle systems with singular interaction through hitting times: application in systemic risk modeling. Ann Appl Probab 29(1), 89–129 (2019)CrossRefGoogle Scholar
  36. Sarantsev, A.: Explicit rates of exponential convergence for reflected jump-diffusions on the half-line. ALEA Lat. Am J Probab Math Stat 13(2), 1069–1093 (2016)Google Scholar
  37. Sarantsev, A.: Reflected Brownian motion in a convex polyhedral cone: tail estimates for the stationary distribution. J Theor Probab 30(3), 1200–1223 (2017)CrossRefGoogle Scholar
  38. Sawyer, S.A.: A formula for semigroups, with an application to branching diffusion processes. Trans Am Math Soc 152(1), 1–38 (1970)CrossRefGoogle Scholar
  39. Strong, W., Fouque, J.-P.: Diversity and arbitrage in a regulatory breakup model. Ann Finance 7(3), 349–374 (2011)CrossRefGoogle Scholar
  40. Spiliopoulos, K., Sirignano, J.A., Giesecke, K.: Fluctuation analysis for the loss from default. Stoch Proc Appl 124(7), 2322–2362 (2014)CrossRefGoogle Scholar
  41. Sun, L.-H.: Systemic risk and interbank lending. J Optim Theory Appl 179(2), 400–424 (2018)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Statistics and Applied ProbabilityUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Department of Mathematics and StatisticsUniversity of NevadaRenoUSA

Personalised recommendations