Time-Inconsistent Stochastic Linear–Quadratic Control: Characterization and Uniqueness of Equilibrium
27 Pages Posted: 5 Apr 2015 Last revised: 5 May 2015
Date Written: April 29, 2015
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.
Keywords: time-inconsistency, stochastic linear-quadratic control, uniqueness of equilibrium control, forward-backward stochastic differential equation, mean-variance portfolio selection
JEL Classification: G11, C73, C68, D81
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