Mixed Equilibrium Solution of Time-Inconsistent Mean-Field Stochastic LQ Problem

Mixed (time-consistent) equilibrium solution is proposed in this paper for the time-inconsistent mean-field stochastic linear-quadratic optimal control. In Example 2 of Section 5, for all the time-state initial pairs the mixed equilibrium solution must exist and ten of such solutions can be constructed, while it is shown that the open-loop equilibrium control and the feedback equilibrium strategy must not exist for some initial pairs. Therefore, it is necessary to study the mixed equilibrium solution, which will give us more flexibility to deal with the time-inconsistent optimal control. A mixed equilibrium solution contains two different parts: the pure-feedback-strategy part and the open-loop-control part, which together constitute a time-consistent solution. It is shown that the open-loop-control part will be of the feedback form of the equilibrium state. If we let the pure-feedback-strategy part or the feedback gain of open-loop-control part be zero, then the mixed equilibrium solution will reduce to the standard open-loop equilibrium control and the feedback equilibrium strategy, respectively, both of which have been extensively studied in existing literature. The fact that a pair of feedback strategy and open-loop control is a mixed equilibrium solution is fully characterized via the properties of solutions of three sets of difference equations. The three sets of difference equations are derived to characterize the convexity and stationary condition, which are obtained via the maximum-principle-like methodology together with the forward-backward stochastic difference equations. Furthermore, the cases with a fixed initial pair and all the initial pairs are separately investigated. By applying the derived theory, the multi-period mean-variance portfolio selection is thoroughly considered, and neat condition has been obtained to ensure the existence of equilibrium solutions.