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 [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 [6] 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 ∗IRMAR, Université Rennes 1, 35042 Rennes Cedex, France. The research of this author was partially supported by the Marie Curie ITN grant, “Controlled Systems,” GA 213841/2008. †Mathematical Institute and Nomura Centre for Mathematical Finance, and Oxford–Man Institute of Quantitative Finance, The University of Oxford, Oxford OX2 6GG, UK. The research of this author was partially supported by research grants from the Nomura Centre for Mathematical Finance and the Oxford–Man Institute of Quantitative Finance. ‡Mathematical Institute and Nomura Centre for Mathematical Finance, and Oxford–Man Institute of Quantitative Finance, The University of Oxford, Oxford OX2 6GG, UK. The research of this author was supported by a start-up fund of the University of Oxford, and research grants from the Nomura Centre for Mathematical Finance and the Oxford–Man Institute of Quantitative Finance.