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Zero-Sum Stochastic Differential Game in Finite Horizon Involving Impulse Controls

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Abstract

This paper considers the problem of two-player zero-sum stochastic differential game with both players adopting impulse controls in finite horizon under rather weak assumptions on the cost functions (c and \(\chi \) not decreasing in time). We use the dynamic programming principle and viscosity solutions approach to show existence and uniqueness of a solution for the Hamilton–Jacobi–Bellman–Isaacs (HJBI) partial differential equation (PDE) of the game. We prove that the upper and lower value functions coincide.

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Acknowledgements

The authors thank gratefully Professor A. Popier and Professor S. Hamadène for helpful discussions and suggestions related to this work. We also would like to thank the referees for their careful reading and for their helpful comments and suggestions that led to considerable improvements in the paper. \(\square \)

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Authors and Affiliations

  1. Université Ibn Zohr, Equipe, Aide á la decision, ENSA, B.P. 1136, Agadir, Morocco

    Brahim El Asri & Sehail Mazid

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  1. Brahim El Asri
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  2. Sehail Mazid
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Correspondence to Brahim El Asri.

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El Asri, B., Mazid, S. Zero-Sum Stochastic Differential Game in Finite Horizon Involving Impulse Controls. Appl Math Optim 81, 1055–1087 (2020). https://doi.org/10.1007/s00245-018-9529-2

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