Time-inconsistent stochastic linear quadratic control for discrete-time systems

Abstract

This paper is mainly concerned with the time-inconsistent stochastic linear quadratic (LQ) control problem in a more general formulation for discrete-time systems. The time-inconsistency arises from three aspects: the coefficient matrices depending on the initial pair, the terminal of the cost function involving the initial pair together with the nonlinear terms of the conditional expectation. The main contributions are: firstly, the maximum principle is derived by using variational methods, which forms a flow of forward and backward stochastic difference equations (FBSDE); secondly, in the case of the system state being one-dimensional, the equilibrium control is obtained by solving the FBSDE with feedback gain based on several nonsymmetric Riccati equations; finally, the necessary and sufficient solvability condition for the time-inconsistent LQ control problem is presented explicitly. The key techniques adopted are the maximum principle and the solution to the FBSDE developed in this paper.

Closed-loop Equilibrium for Time-Inconsistent McKean-Vlasov Controlled Problem

“…Lots of works have been devoted to dealing with time-inconsistency in the last decade (e.g. see [7,2,4,3,27,32,33,22,30]). One also may refer to the survey paper [31] and the references therein.…”

Mixed Equilibrium Solution of Time-Inconsistent Stochastic Linear-Quadratic Problem

“…Yong further developed Strotz's equilibrium solution [24], which is essentially a closed-loop equilibrium strategy, into the LQ optimal control [27,31] and the nonlinear optimal control [29,28,26]. The open-loop equilibrium control has been extensively studied by Hu-Jin-Zhou [12,13], Yong [31], Ni-Zhang-Krstic [18], and Qi-Zhang [22]. In particularly, the closed-loop formulation can be viewed as an extension of Bellman's dynamic programming, and the corresponding equilibrium strategy (if it exists) is constructed by a backward procedure [27,28,29,31].…”

Equilibrium Solutions of Multiperiod Mean-Variance Portfolio Selection

“…It is shown that the open-loop-control part will be of the feedback form of the equilibrium state. If we let the pure-feedbackstrategy part be zero or let the open-loop-control part be independent of the initial state, then the mixed equilibrium solution will reduce to the open-loop equilibrium control and the (linear) feedback equilibrium strategy, respectively, both of which have been extensively studied in existing literature [1,2,4,5,6,7,11,12,16,18,19]. Furthermore, the mixed equilibrium solution is not a hollow concept, whose study will give us more flexibility to deal with the time-inconsistent optimal control.The multi-period mean-variance portfolio selection is a particular example of time-inconsistent problem.…”

“…In recent years, there is a surge to study the time-inconsistent optimal control together with the revisit to multi-period mean-variance portfolio selection [1,2,4,5,6,7,11,12,16,18,19]. Two kinds of time-consistent equilibrium solutions are investigated in these papers, which are the open-loop equilibrium control and the closed-loop equilibrium strategy.…”

“…Strotz's equilibrium solution [14] is essentially a closed-loop equilibrium strategy, which is further elaborately developed by Yong to the LQ optimal control [16,19] as well as the nonlinear optimal control [18,17,15]. In contrast, open-loop equilibrium control is extensively studied by Hu-Jin-Zhou [6,7], Yong [19], Ni-Zhang-Krstic [11], and Qi-Zhang [12]. In particular, the closed-loop formulation can be viewed as the extension of Bellman's dynamic programming, and the corresponding equilibrium strategy (if it exists) is derived by a backward procedure [16,17,18,19].…”