A Zero-Sum Stochastic Differential Game with Impulses, Precommitment, and Unrestricted Cost Functions

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Abstract

We study a zero-sum stochastic differential game (SDG) in which one controller plays an impulse control while their opponent plays a stochastic control. We consider an asymmetric setting in which the impulse player commits to, at the start of the game, performing less than q impulses (q can be chosen arbitrarily large). In order to obtain the uniform continuity of the value functions, previous works involving SDGs with impulses assume the cost of an impulse to be decreasing in time. Our work avoids such restrictions by requiring impulses to occur at rational times. We establish that the resulting game admits a value, and in turn, the existence and uniqueness of viscosity solutions to an associated Hamilton-Jacobi-Bellman-Isaacs quasi-variational inequality.

Keywords

Zero-sum stochastic differential game Impulse control Quasi-variational inequalities Viscosity solutions 

Mathematics Subject Classification

49L20 49L25 91A23 91A15 

Notes

Acknowledgements

The author wishes to express his deepest gratitude to Catherine Rainer (Université de Brest) for all of her help, support, and for pointing out the use of delay strategies to establish the pointwise inequality \(v^{-}\leqslant v^{+}\). The author also thanks Andrea Cosso (Politecnico di Milano) for discussion regarding DPPs, Christine Grün (Université Toulouse 1 Capitole) for discussion on games, and Nizar Touzi (École Polytechnique) for discussion regarding comparison.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

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