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

27 Pages Posted: 5 Apr 2015 Last revised: 5 May 2015

See all articles by Ying Hu

Ying Hu

Université de Rennes 1

Hanqing Jin

Mathematical Institute; Oxford-Nie Financial Big Data Laboratory; St. Peter's College

Xun Yu Zhou

Columbia University - Department of Industrial Engineering and Operations Research (IEOR)

Date Written: April 29, 2015

Abstract

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

Suggested Citation

Hu, Ying and Jin, Hanqing and Zhou, Xunyu, Time-Inconsistent Stochastic Linear–Quadratic Control: Characterization and Uniqueness of Equilibrium (April 29, 2015). Available at SSRN: https://ssrn.com/abstract=2589518 or http://dx.doi.org/10.2139/ssrn.2589518

Ying Hu

Université de Rennes 1 ( email )

11 Rue Jean Macé
Rennes, Rennes 35042
France

Hanqing Jin (Contact Author)

Mathematical Institute ( email )

Andrew Wiles Building
Radicliff Observatory Quarter, Woodstock Road
Oxford, oxfordshire OX2 6GG
United Kingdom

HOME PAGE: http://www.maths.ox.ac.uk

Oxford-Nie Financial Big Data Laboratory ( email )

Andrew Wiles Building
Woodstock Road
Oxford, Oxfordshire OX2 6GG
United Kingdom

St. Peter's College ( email )

New Inn Hall Street
Oxford, Oxfordshire OX1 2DL
United Kingdom

HOME PAGE: http://www.spc.ox.ac.uk

Xunyu Zhou

Columbia University - Department of Industrial Engineering and Operations Research (IEOR) ( email )

331 S.W. Mudd Building
500 West 120th Street
New York, NY 10027
United States

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