朱庆峰,王天啸,石玉峰.平均场倒向重随机微分方程及其应用[J].数学年刊A辑,2020,(4):409~428
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
Received:November 17, 2018  Revised:August 10, 2020
DOI:10.16205/j.cnki.cama.2020.0029
中文关键词:  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
英文关键词:Mean-Field, Backward doubly stochastic differential equations,Nonlocal stochastic partial differential equations, Maximum principle, Linear quadratic optimal control
基金项目: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)和泰山学者工程专项经费(No.tsqn20161041)
Author NameAffiliation
ZHU Qingfeng 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. 
WANG Tianxiao School of Mathematics, Sichuan University, Chengdu 610065, China. 
SHI Yufeng Corresponding author. Institute for Financial Studies and School of Mathematics, Shandong University, Jinan 250100, China. 
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中文摘要:
      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.
英文摘要:
      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.
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