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

Citation:

Maoning TANG,Qingxin MENG,Meijiao WANG.Forward and Backward Mean-Field Stochastic Partial Differential Equation and Optimal Control[J].Chinese Annals of Mathematics B,2019,40(4):515~540
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Authors:

Maoning TANG; Qingxin MENG;Meijiao WANG

Foundation:

This work was supported by the National Natural Science Foundation of China (Nos.11871121, 11471079, 11301177) and the Natural Science Foundation of Zhejiang Province for Distinguished Young Scholar (No.LR15A010001).
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, Backwardstochastic partial differential equation, Optimal control, Maximumprinciple, Adjoint equation

Classification:

60H15, 35R60, 93E20
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