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

Ying Hu (1) , Hanqing Jin (2) , Xun Yu Zhou (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 University of Oxford

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.

Mots clés: time-inconsistency; mean--variance portfolio selection; stochastic linear-quadratic control; uniqueness of equilibrium control; forward--backward stochastic differential equation

Domaines: Gestion de portefeuilles [q-fin.PM]; Probabilités [math.PR]

Ying Hu, Hanqing Jin, Xun Yu Zhou. Time-Inconsistent Stochastic Linear--Quadratic Control: Characterization and Uniqueness of Equilibrium. SIAM Journal on Control and Optimization, 2017, 55 (2), pp.1261-1279. ⟨10.1137/15M1019040⟩. ⟨hal-01139343v2⟩