Time-Inconsistent Stochastic Linear--Quadratic Control: Characterization and Uniqueness of Equilibrium

TL;DR

This paper characterizes and proves the uniqueness of equilibrium controls in time-inconsistent stochastic linear-quadratic problems, including applications to mean-variance portfolio optimization in stochastic markets.

Contribution

It provides a necessary and sufficient condition for equilibrium controls and establishes their uniqueness in specific deterministic and stochastic settings.

Findings

Explicit equilibrium control is unique in one-dimensional deterministic coefficient cases.

Equilibrium strategy is unique in mean-variance portfolio models with stochastic market parameters.

The paper introduces a stochastic Lebesgue differentiation theorem for analyzing equilibria.

Abstract

In this paper, we continue our study on a general time-inconsistent stochastic linear--quadratic (LQ) control problem originally formulated in [6]. We derive a necessary and 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 prove that the explicit equilibrium control constructed in \cite{HJZ} is indeed unique. Our proof is based on the derived equivalent condition for equilibria as well as a stochastic version of the Lebesgue differentiation theorem. Finally, we show that the equilibrium strategy is unique for 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.