Time-Inconsistent Stochastic Linear--Quadratic Control

• Published in: SIAM journal on control and optimization Vol. 50; no. 3; pp. 1548 - 1572
• Main Authors: Hu, Ying, Jin, Hanqing, Zhou, Xun Yu
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2012
• Subjects: Applied mathematics; Dynamical Systems; Equilibrium; Mathematics; Optimization and Control; Portfolio Management; Probability; Quantitative Finance; Random variables; Stochastic control theory; time-inconsistency; forward-backward stochastic differential equation; mean-variance portfolio selection; stochastic linear-quadratic control; equilibriumcontrol
• ISSN: 0363-0129; 1095-7138
• DOI: 10.1137/110853960

In this paper, we formulate a general time-inconsistent stochastic linear--quadratic (LQ) control problem. The time-inconsistency arises from the presence of a quadratic term of the expected state as well as a state-dependent term in the objective functional. We define an equilibrium, instead of optimal, solution within the class of open-loop controls, and derive a sufficient condition for equilibrium controls via a flow of forward--backward stochastic differential equations. When the state is one dimensional and the coefficients in the problem are all deterministic, we find an explicit equilibrium control. As an application, we then consider a mean--variance portfolio selection model in a complete financial market where the risk-free rate is a deterministic function of time but all the other market parameters are possibly stochastic processes. Applying the general sufficient condition, we obtain explicit equilibrium strategies when the risk premium is both deterministic and stochastic. [PUBLICATION ABSTRACT]