平均场倒向重随机微分方程及其应用  

Mean-Field Backward Doubly Stochastic Differential Equations and Its Applications

作　　者：朱庆峰 王天啸[3] 石玉峰 ZHU Qingfeng;WANG Tianxiao;SHI Yufeng(School of Mathematics and Quantitative Economics,Shandong University of Finance and Economics,Shandong Key Laboratory of Blockchain Finance,Jinan 250014,China;Institute for Financial Studies and School of Mathematics,Shandong University,Jinan 250100,China;School of Mathematics,Sichuan University,Chengdu 610065,China)

机构地区：[1]山东财经大学数学与数量经济学院,山东省区块链金融重点实验室,济南250014 [2]山东大学金融研究院和数学学院,济南250100 [3]四川大学数学学院,成都610065

出　　处：《数学年刊（A辑）》2020年第4期409-428,共20页Chinese Annals of Mathematics

基　　金：国家重点研发计划(No.2018YFA0703900);国家自然科学基金(No.11871309,No.11671229,No.11971332,No.11931011,No.71871129,No.11371226,No.11301298);山东省自然科学基金(No.ZR2019MA013);山东省高等教育科技计划项目(No.J17KA162);泰山学者工程专项经费(No.tsqn20161041)项目的资助。

摘　　要：研究了平均场倒向随重机微分方程,得到了平均场倒向重随机微分方程解的存在唯一性.基于平均场倒向重随机微分方程的解,给出了一类非局部随机偏微分方程解的概率解释.讨论了平均场倒向重随机系统的最优控制问题,建立了庞特利亚金型的最大值原理.最后讨论了一个平均场倒向重随机线性二次最优控制问题,展示了上述最大值原理的应用.In this paper,mean-field backward doubly stochastic differential equations(MFBDSDEs in short)are introduced and studied.The existence and uniqueness of solutions for MF-BDSDEs is established.One probabilistic interpretation for the solutions to a class of nonlocal stochastic partial differential equations is given.A maximum principle of Pontryagin’s type is established for optimal control problems of MF-BDSDEs.Finally,one backward linear quadratic optimal control problem of mean-field type is discussed to illustrate the direct application of the maximum principle.

关 键 词：平均场 倒向重随机微分方程 非局部随机偏微分方程 最大值原理 线性二次最优控制 

分 类 号：O211.6[理学—概率论与数理统计]