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Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 718948, 18 pages
http://dx.doi.org/10.1155/2014/718948
Research Article

Mean-Field Backward Stochastic Evolution Equations in Hilbert Spaces and Optimal Control for BSPDEs

1School of Mathematics, Shandong University, Jinan 250100, China
2School of Mathematics, Shandong Polytechnic University, Jinan 250353, China
3School of Mathematical Sciences, Shandong Normal University, Jinan 250014, China

Received 31 March 2014; Revised 23 May 2014; Accepted 18 June 2014; Published 13 July 2014

Academic Editor: Guangchen Wang

Copyright © 2014 Ruimin Xu and Tingting Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We obtain the existence and uniqueness result of the mild solutions to mean-field backward stochastic evolution equations (BSEEs) in Hilbert spaces under a weaker condition than the Lipschitz one. As an intermediate step, the existence and uniqueness result for the mild solutions of mean-field BSEEs under Lipschitz condition is also established. And then a maximum principle for optimal control problems governed by backward stochastic partial differential equations (BSPDEs) of mean-field type is presented. In this control system, the control domain need not to be convex and the coefficients, both in the state equation and in the cost functional, depend on the law of the BSPDE as well as the state and the control. Finally, a linear-quadratic optimal control problem is given to explain our theoretical results.