A zero-sum stochastic differential game with impulses, precommitment, and unrestricted cost functions

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 using a dynamic programming principle with finite-valued stopping 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.

arXiv commentCorrected DPP/DPE, comparison, and control-reduction proofs; removed the rational-time restriction on impulses; added conditions for measurable strategy pasting. Value, comparison and DPE conclusions unchanged. Minor bibliography/typesetting updates

References in corpus (4)

• Stochastic Differential Games and Viscosity Solutions of Hamilton-Jacobi-Bellman-Isaacs Equations
• Backward SDEs with constrained jumps and quasi-variational inequalities
• Optimal stopping under adverse nonlinear expectation and related games
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Cited by in corpus (4)

• Feedback Nash Equilibria in Differential Games with Impulse Control
• Zero-sum stochastic differential game in finite horizon involving impulse controls
• A fixed-point policy-iteration-type algorithm for symmetric nonzero-sum stochastic impulse control games
• Continuous and Impulse Controls Differential Game in Finite Horizon with Nash-Equilibrium and Application