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Equilibrium for a Time-Inconsistent Stochastic Linear–Quadratic Control System with Jumps and Its Application to the Mean-Variance Problem

  • Zhongyang SunEmail author
  • Xianping Guo
Article
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Abstract

This paper studies a kind of time-inconsistent linear–quadratic control problem in a more general framework with stochastic coefficients and random jumps. The time inconsistency comes from the dependence of the terminal cost on the current state as well as the presence of a quadratic term of the expected terminal state in the objective functional. Instead of finding a global optimal control, we look for a time-consistent locally optimal equilibrium solution within the class of open-loop controls. A general sufficient and necessary condition for equilibrium controls via a flow of forward–backward stochastic differential equations is derived. This paper further develops a new methodology to cope with the mathematical difficulties arising from the presence of stochastic coefficients and random jumps. As an application, we study a mean-variance portfolio selection problem in a jump-diffusion financial market; an explicit equilibrium investment strategy in a deterministic coefficients case is obtained and proved to be unique.

Keywords

Time-inconsistent linear–quadratic control Stochastic coefficients and random jumps Equilibrium control Forward–backward stochastic differential equation Mean-variance portfolio selection 

Mathematics Subject Classification

91G80 93E20 

Notes

Acknowledgements

The authors would like to thank the referees for their careful reading of the paper and helpful suggestions. This work was supported by the National Natural Science Foundation of China (NSFC Grant Nos. 11571189, 11701087, 61773411) and Shandong Provincial Natural Science Foundation, China.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of StatisticsQufu Normal UniversityQufuPeople’s Republic of China
  2. 2.School of MathematicsSun Yat-sen UniversityGuangzhouPeople’s Republic of China

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