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 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.

Nonzero-Sum Stochastic Differential Games with Impulse Controls: A Verification Theorem with Applications

“…Regarding stochastic games with impulse controls, as well as games with mixed impulse and continuous controls, the existing literature almost entirely focuses on the zero-sum case. We refer to [3,26] for zero-sum games where one player uses continuous controls and the opponent takes impulse controls. For zero-sum games where both players use impulse controls, we cite [25] for the case with delay and the recent works [16,19] for a viscosity approach.…”

Nonzero-Sum Stochastic Differential Games Between an Impulse Controller and a Stopper

“…Regarding the zero-sum case, here we quote Cosso [23], who examined a finite time horizon two-player game where both players act via impulse control strategies and showed that such games have a value which is the unique viscosity solution of the double-obstacle quasivariational inequality. Furthermore, Azimzadeh [24] considered an asymmetric setting with one participant playing a regular control, while the opponent is playing an impulse control with precommit-ment, meaning that at the beginning of the game the maximum number of impulses is declared, and proved that such a game has a value in the viscosity sense.…”

A Fixed-Point Policy-Iteration-Type Algorithm for Symmetric Nonzero-Sum Stochastic Impulse Control Games

“…It is not surprising then, that the literature in impulse games is limited and mainly focused on the zero-sum case [8,17,21]. The more general and versatile nonzero-sum instance has only recently received attention.…”

“…More precisely, these would be R-admissible strategies. See [1, Def.2.5] for more details 8. Although we could have M i V i (x) = +∞, this will be excluded when enforcing the system of QVIs (1.2.3) 9.…”