## Abstract

In this article, we study mean-field-type games with jump–diffusion and regime switching in which the payoffs and the state dynamics depend not only on the state–action profile of the decision-makers but also on a measure of the state–action pair. The state dynamics is a measure-dependent process with jump–diffusion and regime switching. We derive novel equilibrium systems to be solved. Two solution approaches are presented: (i) dynamic programming principle and (ii) stochastic maximum principle. Relationship between dual function and adjoint processes are provided. It is shown that the extension to the risk-sensitive case generates a nonlinearity to the adjoint process and it involves three other processes associated with the diffusion, jump and regime switching, respectively.

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## Acknowledgements

Alain Bensoussan is supported by Grants from the National Science Foundation NSF-DMS 1612880 and the Research Grants Council of the Hong Kong Special Administrative Region City U-11303316. The research of Boualem Djehiche is supported by Grants from the Swedish Research Council 2016-04086. The research of Hamidou Tembine is supported by US Air Force Office of Scientific Research under Grant Number FA9550-17-1-0259 with title: Mean-Field-Type Games. Sheung Chi Phillip Yam acknowledges the financial support from HKGRF-14300717 with the project title: New Kinds of Forward–Backward Stochastic Systems with Applications, HKSAR-GRF-14301015 with title: Advance in Mean Field Theory, Direct Grant for Research 2014/15 with project code: 4053141 offered by CUHK.

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Bensoussan, A., Djehiche, B., Tembine, H. *et al.* Mean-Field-Type Games with Jump and Regime Switching.
*Dyn Games Appl* **10, **19–57 (2020). https://doi.org/10.1007/s13235-019-00306-2

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### Keywords

- Mean-field
- McKean–Vlasov
- Game theory