Time-Inconsistent Stochastic LinearrQuadratic Control: Characterization and Uniqueness of Equilibrium

Abstract

In this paper, we continue our study on a general time-inconsistent stochastic linear-quadratic (LQ) control problem originally formulated in Hu, Jin and Zhou (2012). 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 Hu, Jin and Zhou (2012) 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.