Systemic Risk and Stochastic Games with Delay

Rene A. Carmona, Jean Pierre Fouque, Seyyed Mostafa Mousavi, Li Hsien Sun

Research output: Contribution to journalArticlepeer-review

21 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)366-399
Number of pages34
JournalJournal of Optimization Theory and Applications
Issue number2
StatePublished - Nov 1 2018

All Science Journal Classification (ASJC) codes

  • Management Science and Operations Research
  • Control and Optimization
  • Applied Mathematics


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


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