A deterministic linear quadratic time-inconsistentoptimal control problem

Jiongmin Yong

doi:10.3934/mcrf.2011.1.83

Article Contents

2011, Volume 1, Issue 1: 83-118. Doi: 10.3934/mcrf.2011.1.83

This issue Previous Article Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain Next Article Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential

A deterministic linear quadratic time-inconsistent optimal control problem

Jiongmin Yong^1,

1.
Department of Mathematics, University of Central Florida, Orlando, FL 32816

Received: October 2010

Revised: November 2010

Published: March 2011

Abstract Related Papers Cited by

Abstract

A time-inconsistent optimal control problem is formulated and studied for a controlled linear ordinary differential equation with a quadratic cost functional. A notion of time-consistent equilibrium strategy is introduced for the original time-inconsistent problem. Under certain conditions, we construct an equilibrium strategy which can be represented via a Riccati--Volterra integral equation system. Our approach is based on a study of multi-person hierarchical differential games.

Keywords:

Mathematics Subject Classification: Primary: 49L20, 49N10, 49N70; Secondary: 91A23, 91A65.

Citation:

Related Papers

Cited by

References

[1]	S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Preprint, London Business School, 2008.
[2]	L. D. Berkovitz, "Optimal Control Theory," Applied Mathematical Sciences, Vol. 12, Springer-Verlag, New York-Heidelberg, 1974.
[3]	T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochasitc control problem, working paper.
[4]	E. V. Böhm-Bawerk, "The Positive Theory of Capital," Books for Libraries Press, Freeport, New York, 1891.
[5]	I. Ekeland and A. Lazrak, Being serious about non-commitment: subgame perfect equilibrium in continuous time, preprint, Univ. British Columbia, 2008.
[6]	I. Ekeland and T. Privu, Investment and consumption without commitment, preprint, Univ. British Columbia, 2007.
[7]	S. M. Goldman, Consistent plans, Review of Economic Studies, 47 (1980), 533-537.doi: 10.2307/2297304.
[8]	S. R. Grenadier and N. Wang, Investment under uncertainty and time-inconsistent preferences, preprint.
[9]	P. J. Herings and K. I. M. Rohde, Time-inconsistent preferences in a general equilibriub model, preprint.
[10]	D. Hume, "A Treatise of Human Nature," First Edition, 1739; Reprint, Oxford Univ. Press, New York, 1978.
[11]	W. S. Jevons, "Theory of Political Economy," Mcmillan, London, 1871.
[12]	P. Krusell and A. A. Smith, Jr., Consumption and saving decisions with quasi-geometric discounting, Econometrica, 71 (2003), 366-375.doi: 10.1111/1468-0262.00400.
[13]	D. Laibson, Golden eggs and hyperbolic discounting, Quarterly J. Econ., 112 (1997), 443-477.doi: 10.1162/003355397555253.
[14]	A. Malthus, An essay on the principle of population, 1826, "The Works of Thomas Robert Malthus" (Eds. E. A. Wrigley and D. Souden), 2 and 3, W. Pickering, London, 1986.
[15]	J. Marin-Solano and J. Navas, Non-constant discounting in finite horizon: The free terminal time case, J. Economic Dynamics and Control, 33 (2009), 666-675.doi: 10.1016/j.jedc.2008.08.008.
[16]	A. Marshall, "Principles of Economics," 1st ed., 1890; 8th ed., Macmillan, London, 1920.
[17]	M. Miller and M. Salmon, Dynamic games and the time inconsistency of optimal policy in open economics, The Economic Journal, 95 (1985), 124-137.doi: 10.2307/2232876.
[18]	I. Palacios-Huerta, Time-inconsistent preferences in Adam Smith and Davis Hume, History of Political Economy, 35 (2003), 241-268.doi: 10.1215/00182702-35-2-241.
[19]	V. Pareto, "Manuel d'économie Politique," Girard and Brieve, Paris, 1909.
[20]	B. Peleg and M. E. Yaari, On the existence of a consistent course of action when tastes are changing, Review of Economic Studies, 40 (1973), 391-401.doi: 10.2307/2296458.
[21]	R. A. Pollak, Consistent planning, Review of Economic Studies, 35 (1968), 185-199.doi: 10.2307/2296548.
[22]	A. Smith, "The Theory of Moral Sentiments," First Edition, 1759; Reprint, Oxford Univ. Press, 1976.
[23]	R. H. Strotz, Myopia and inconsistency in dynamic utility maximization, Review of Econ. Studies, 23 (1955), 165-180.doi: 10.2307/2295722.
[24]	L. Tesfatsion, Time inconsistency of benevolent government economics, J. Public Economics, 31 (1986), 25-52.doi: 10.1016/0047-2727(86)90070-8.
[25]	J. Yong, A deterministic time-inconsistent optimal control problem -- An essentially cooperative approach, Acta Appl. Math. Sinica, to appear.
[26]	J. Yong, and X. Y. Zhou, "Stochastic Controls: Hamiltonian Systems and HJB Equations," Applications of Mathematics (New York), 43, Springer-Verlag, New York, 1999.