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Chinese Annals of Mathematics, Series B

, Volume 40, Issue 4, pp 515–540 | Cite as

Forward and Backward Mean-Field Stochastic Partial Differential Equation and Optimal Control

  • Maoning Tang
  • Qingxin MengEmail author
  • Meijiao Wang
Article

Abstract

This paper is mainly concerned with the solutions to both forward and backward mean-field stochastic partial differential equation and the corresponding optimal control problem for mean-field stochastic partial differential equation. The authors first prove the continuous dependence theorems of forward and backward mean-field stochastic partial differential equations and show the existence and uniqueness of solutions to them. Then they establish necessary and sufficient optimality conditions of the control problem in the form of Pontryagin’s maximum principles. To illustrate the theoretical results, the authors apply stochastic maximum principles to study the infinite-dimensional linear-quadratic control problem of mean-field type. Further, an application to a Cauchy problem for a controlled stochastic linear PDE of mean-field type is studied.

Keywords

Mean-field Stochastic partial differential equation Backward stochastic partial differential equation Optimal control Maximum principle Adjoint equation 

2000 MR Subject Classification

60H15 35R60 93E20 

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Copyright information

© The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesHuzhou UniversityHuzhou, ZhejiangChina
  2. 2.Business SchoolUniversity of Shanghai for Science and TechnologyShanghaiChina

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