Time-inconsistent stochastic linear-quadratic control: characterization and uniqueness of equilibrium

In this paper, we continue our study on a general time-inconsistent stochastic linear– quadratic (LQ) control problem originally formulated in [9]. 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 [9] 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.