Mixed Equilibrium Solution of Time-Inconsistent Stochastic LQ Problem

TL;DR

This paper introduces a new mixed equilibrium concept for time-inconsistent stochastic LQ problems, combining feedback strategies and open-loop controls, with conditions for existence and examples demonstrating its broader applicability.

Contribution

It proposes the mixed equilibrium solution, unifying open-loop and feedback approaches, and provides necessary and sufficient conditions for its existence using stochastic difference equations.

Findings

Mixed equilibrium solutions can exist even when traditional solutions do not.

The paper establishes a maximum-principle-like condition for existence.

An example demonstrates the broader applicability of the mixed solution concept.

Abstract

In this paper, we propose a novel equilibrium solution notion for the time-inconsistent stochastic linear-quadratic optimal control problem. This notion is called the mixed equilibrium solution, which consists of two parts: a pure-feedback-strategy part and an open-loop-control part. When the pure-feedback-strategy part is zero or the open-loop-control part does not depend on the initial state, the mixed equilibrium solution reduces to the open-loop equilibrium control and the feedback equilibrium strategy, respectively. Using a maximum-principle-like methodology with forward-backward stochastic difference equations, a necessary and sufficient condition is established to characterize the existence of a mixed equilibrium solution. Then, by decoupling the forward-backward stochastic difference equations, three sets of difference equations, which together portray the existence of a mixed equilibrium solution, are obtained. Moreover, the case with a fixed time-state initial pair and the case with all the initial pairs are separately investigated. Furthermore, an example is constructed to show that the mixed equilibrium solution exists for all the initial pairs, although neither the open-loop equilibrium control nor the feedback equilibrium strategy exists for some initial pairs.