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Mean field games with controlled jump–diffusion dynamics: Existence results and an illiquid interbank market model

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  • Benazzoli, Chiara
  • Campi, Luciano
  • Di Persio, Luca

Abstract

We study a family of mean field games with a state variable evolving as a multivariate jump–diffusion process. The jump component is driven by a Poisson process with a time-dependent intensity function. All coefficients, i.e. drift, volatility and jump size, are controlled. Under fairly general conditions, we establish existence of a solution in a relaxed version of the mean field game and give conditions under which the optimal strategies are in fact Markovian, hence extending to a jump–diffusion setting previous results established in Lacker (2015). The proofs rely upon the notions of relaxed controls and martingale problems. Finally, to complement the abstract existence results, we study a simple illiquid inter-bank market model, where the banks can change their reserves only at the jump times of some exogenous Poisson processes with a common constant intensity, and provide some numerical results.

Suggested Citation

  • Benazzoli, Chiara & Campi, Luciano & Di Persio, Luca, 2020. "Mean field games with controlled jump–diffusion dynamics: Existence results and an illiquid interbank market model," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 6927-6964.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:11:p:6927-6964
    DOI: 10.1016/j.spa.2020.07.004
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    References listed on IDEAS

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    1. Fu, Guanxing & Horst, Ulrich, 2017. "Mean Field Games with Singular Controls," Rationality and Competition Discussion Paper Series 22, CRC TRR 190 Rationality and Competition.
    2. Olivier Guéant & Pierre Louis Lions & Jean-Michel Lasry, 2011. "Mean Field Games and Applications," Post-Print hal-01393103, HAL.
    3. Lacker, Daniel, 2015. "Mean field games via controlled martingale problems: Existence of Markovian equilibria," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2856-2894.
    4. Graham, Carl, 1992. "McKean-Vlasov Ito-Skorohod equations, and nonlinear diffusions with discrete jump sets," Stochastic Processes and their Applications, Elsevier, vol. 40(1), pages 69-82, February.
    5. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, December.
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    Cited by:

    1. Wu, Mingyan & Hao, Zimo, 2023. "Well-posedness of density dependent SDE driven by α-stable process with Hölder drifts," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 416-442.
    2. Andrés Cárdenas & Sergio Pulido & Rafael Serrano, 2022. "Existence of optimal controls for stochastic Volterra equations," Working Papers hal-03720342, HAL.
    3. Lijun Bo & Shihua Wang & Xiang Yu, 2021. "Mean Field Game of Optimal Relative Investment with Jump Risk," Papers 2108.00799, arXiv.org, revised Feb 2023.

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