Optimal control of mean-field backward doubly stochastic systems driven by Itô-Lévy processes

ABSTRACT

In this paper, we introduce a new class of backward doubly stochastic differential equations (in short BDSDE) called mean-field backward doubly stochastic differential equations (in short MFBDSDE) driven by Itô-Lévy processes and study the partial information optimal control problems for backward doubly stochastic systems driven by Itô-Lévy processes of mean-field type, in which the coefficients depend on not only the solution processes but also their expected values. First, using the method of contraction mapping, we prove the existence and uniqueness of the solutions to this kind of MFBDSDE. Then, by the method of convex variation and duality technique, we establish a sufficient and necessary stochastic maximum principle for the stochastic system. Finally, we illustrate our theoretical results by an application to a stochastic linear quadratic optimal control problem of a mean-field backward doubly stochastic system driven by Itô-Lévy processes.

KEYWORDS:

• Stochastic optimal control
• stochastic maximum principle
• mean-field backward doubly stochastic differential equations driven by Itô-Lévy processes

Acknowledgments

The authors thank the anonymous referees for the helpful suggestions and valuable comments which improved this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is supported by the National Natural Science Foundation of China [grant number 11671404], Provincial Natural Science Foundation of Hunan [grant number 2017JJ3405] and the Yu Ying project of Central South University.