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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

Abstract

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.

Keywords

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 

Notes

Acknowledgements

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.

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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

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