A zero-sum hybrid stochastic differential game with impulse controls  

作　　者：Siyu LV Zhen WU Jie XIONG 

机构地区：[1]School of Mathematics,Southeast University,Nanjing 211189,China [2]School of Mathematics,Shandong University,Jinan 250100,China [3]Department of Mathematics and SUSTech International Center for Mathematics,Southern University of Science and Technology,Shenzhen 518055,China

出　　处：《Science China(Information Sciences)》2024年第11期281-295,共15页中国科学（信息科学）（英文版）

基　　金：supported by National Key R&D Program of China(Grant Nos.2023YFA1009200,2022YFA-1006102);National Natural Science Foundation of China(Grant Nos.12471414,11831010,61961160732,12471418);Natural Science Foundation of Jiangsu Province(Grant No.BK20242023);Natural Science Foundation of Shandong Province(Grant No.ZR2019ZD42);Taishan Scholars Climbing Program of Shandong(Grant No.TSPD20210302);Fundamental Research Funds for the Central Universities(Grant No.2242024K40018);Jiangsu Province Scientific Research Center of Applied Mathematics(Grant No.BK20233002)。

摘　　要：In this paper,we study a zero-sum stochastic differential game with the following salient features:(i)the system state is dictated by a hybrid diffusion,(ii)both players use impulse controls,and(iii)the game takes place on an infinite time horizon.First,the dynamic programming principle for the problem is proven.Then,the lower and upper value functions of the game are characterized as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman-Isaacs(HJBI)equation,which turns out to be a coupled system of variational inequalities with bilateral obstacles.Moreover,a verification theorem as a sufficient condition to identify a Nash equilibrium is established.The Nash equilibrium strategies for the two players,indicating when and how it is optimal to intervene,are given in terms of the obstacle part of the HJBI equation.

关 键 词：stochastic differential game Markov chain impulse control HJBI equation viscosity solution verification theorem 

分 类 号：O225[理学—运筹学与控制论]