Time-Inconsistent Stochastic Linear--Quadratic Control

Ying Hu (), Hanqing Jin and Xun Yu Zhou
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Ying Hu: IRMAR - Institut de Recherche Mathématique de Rennes - UR - Université de Rennes - INSA Rennes - Institut National des Sciences Appliquées - Rennes - INSA - Institut National des Sciences Appliquées - ENS Rennes - École normale supérieure - Rennes - UR2 - Université de Rennes 2 - CNRS - Centre National de la Recherche Scientifique - INSTITUT AGRO Agrocampus Ouest - Institut Agro - Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement

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Abstract: In this paper, we formulate a general time-inconsistent stochastic linear--quadratic (LQ) control problem. The time-inconsistency arises from the presence of a quadratic term of the expected state as well as a state-dependent term in the objective functional. We define an equilibrium, instead of optimal, solution within the class of open-loop controls, and derive a 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 find an explicit equilibrium control. As an application, we then consider 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. Applying the general sufficient condition, we obtain explicit equilibrium strategies when the risk premium is both deterministic and stochastic.

Keywords: time-inconsistency; stochastic linear-quadratic control; equilibriumcontrol; forward-backward stochastic differential equation; mean-variance portfolio selection (search for similar items in EconPapers)
Date: 2012
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Citations: View citations in EconPapers (73)

Published in SIAM Journal on Control and Optimization, 2012, 50 (3), pp.1548-1572. ⟨10.1137/110853960⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-00691816

DOI: 10.1137/110853960

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