Systemic Risk and Stochastic Games with Delay

  • René Carmona
  • Jean-Pierre Fouque
  • Seyyed Mostafa Mousavi
  • Li-Hsien Sun
S.I. : Optimization and Stochastic Control in Finance


We propose a model of inter-bank lending and borrowing which takes into account clearing debt obligations. The evolution of log-monetary reserves of banks is described by coupled diffusions driven by controls with delay in their drifts. Banks are minimizing their finite-horizon objective functions which take into account a quadratic cost for lending or borrowing and a linear incentive to borrow if the reserve is low or lend if the reserve is high relative to the average capitalization of the system. As such, our problem is a finite-player linear–quadratic stochastic differential game with delay. An open-loop Nash equilibrium is obtained using a system of fully coupled forward and advanced-backward stochastic differential equations. We then describe how the delay affects liquidity and systemic risk characterized by a large number of defaults. We also derive a closed-loop Nash equilibrium using a Hamilton–Jacobi–Bellman partial differential equation approach.


Systemic risk Inter-bank borrowing and lending Stochastic game with delay Nash equilibrium 

Mathematics Subject Classification

91A15 91G80 60G99 



The authors would like to thank Romuald Elie and Phillip Yam for conversations on this subject. We also thank an anonymous referee for his/her comments and suggestions which helped make the paper clearer. The work of the Jean-Pierre Fouque was supported by NSF Grant DMS-1409434. The work of the Li-Hsien Sun was supported by MOST-103-2118-M-008-006-MY2.


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Authors and Affiliations

  1. 1.ORFE, Bendheim Center for FinancePrinceton UniversityPrincetonUSA
  2. 2.Department of Statistics and Applied ProbabilityUniversity of California Santa BarbaraSanta BarbaraUSA
  3. 3.Institute of StatisticsNational Central UniversityTaoyuan CityTaiwan

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