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March  2011, 1(1): 83-118. doi: 10.3934/mcrf.2011.1.83

A deterministic linear quadratic time-inconsistent optimal control problem

1. 

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

Received  October 2010 Revised  November 2010 Published  March 2011

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.
Citation: Jiongmin Yong. A deterministic linear quadratic time-inconsistent optimal control problem. Mathematical Control & Related Fields, 2011, 1 (1) : 83-118. doi: 10.3934/mcrf.2011.1.83
References:
[1]

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L. D. Berkovitz, "Optimal Control Theory,", Applied Mathematical Sciences, 12 (1974).   Google Scholar

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S. M. Goldman, Consistent plans,, Review of Economic Studies, 47 (1980), 533.  doi: 10.2307/2297304.  Google Scholar

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S. R. Grenadier and N. Wang, Investment under uncertainty and time-inconsistent preferences,, preprint., ().   Google Scholar

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P. J. Herings and K. I. M. Rohde, Time-inconsistent preferences in a general equilibriub model,, preprint., ().   Google Scholar

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D. Hume, "A Treatise of Human Nature,", First Edition, (1739).   Google Scholar

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W. S. Jevons, "Theory of Political Economy,", Mcmillan, (1871).   Google Scholar

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P. Krusell and A. A. Smith, Jr., Consumption and saving decisions with quasi-geometric discounting,, Econometrica, 71 (2003), 366.  doi: 10.1111/1468-0262.00400.  Google Scholar

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D. Laibson, Golden eggs and hyperbolic discounting,, Quarterly J. Econ., 112 (1997), 443.  doi: 10.1162/003355397555253.  Google Scholar

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A. Malthus, An essay on the principle of population, 1826,, in, 2 (1986).   Google Scholar

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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.  doi: 10.1016/j.jedc.2008.08.008.  Google Scholar

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A. Marshall, "Principles of Economics,", 1st ed., (1890).   Google Scholar

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M. Miller and M. Salmon, Dynamic games and the time inconsistency of optimal policy in open economics,, The Economic Journal, 95 (1985), 124.  doi: 10.2307/2232876.  Google Scholar

[18]

I. Palacios-Huerta, Time-inconsistent preferences in Adam Smith and Davis Hume,, History of Political Economy, 35 (2003), 241.  doi: 10.1215/00182702-35-2-241.  Google Scholar

[19]

V. Pareto, "Manuel d'économie Politique,", Girard and Brieve, (1909).   Google Scholar

[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.  doi: 10.2307/2296458.  Google Scholar

[21]

R. A. Pollak, Consistent planning,, Review of Economic Studies, 35 (1968), 185.  doi: 10.2307/2296548.  Google Scholar

[22]

A. Smith, "The Theory of Moral Sentiments,", First Edition, (1759).   Google Scholar

[23]

R. H. Strotz, Myopia and inconsistency in dynamic utility maximization,, Review of Econ. Studies, 23 (1955), 165.  doi: 10.2307/2295722.  Google Scholar

[24]

L. Tesfatsion, Time inconsistency of benevolent government economics,, J. Public Economics, 31 (1986), 25.  doi: 10.1016/0047-2727(86)90070-8.  Google Scholar

[25]

J. Yong, A deterministic time-inconsistent optimal control problem -- An essentially cooperative approach,, Acta Appl. Math. Sinica, ().   Google Scholar

[26]

J. Yong, and X. Y. Zhou, "Stochastic Controls: Hamiltonian Systems and HJB Equations,", Applications of Mathematics (New York), (1999).   Google Scholar

show all references

References:
[1]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation,, Preprint, (2008).   Google Scholar

[2]

L. D. Berkovitz, "Optimal Control Theory,", Applied Mathematical Sciences, 12 (1974).   Google Scholar

[3]

T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochasitc control problem,, working paper., ().   Google Scholar

[4]

E. V. Böhm-Bawerk, "The Positive Theory of Capital,", Books for Libraries Press, (1891).   Google Scholar

[5]

I. Ekeland and A. Lazrak, Being serious about non-commitment: subgame perfect equilibrium in continuous time,, preprint, (2008).   Google Scholar

[6]

I. Ekeland and T. Privu, Investment and consumption without commitment,, preprint, (2007).   Google Scholar

[7]

S. M. Goldman, Consistent plans,, Review of Economic Studies, 47 (1980), 533.  doi: 10.2307/2297304.  Google Scholar

[8]

S. R. Grenadier and N. Wang, Investment under uncertainty and time-inconsistent preferences,, preprint., ().   Google Scholar

[9]

P. J. Herings and K. I. M. Rohde, Time-inconsistent preferences in a general equilibriub model,, preprint., ().   Google Scholar

[10]

D. Hume, "A Treatise of Human Nature,", First Edition, (1739).   Google Scholar

[11]

W. S. Jevons, "Theory of Political Economy,", Mcmillan, (1871).   Google Scholar

[12]

P. Krusell and A. A. Smith, Jr., Consumption and saving decisions with quasi-geometric discounting,, Econometrica, 71 (2003), 366.  doi: 10.1111/1468-0262.00400.  Google Scholar

[13]

D. Laibson, Golden eggs and hyperbolic discounting,, Quarterly J. Econ., 112 (1997), 443.  doi: 10.1162/003355397555253.  Google Scholar

[14]

A. Malthus, An essay on the principle of population, 1826,, in, 2 (1986).   Google Scholar

[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.  doi: 10.1016/j.jedc.2008.08.008.  Google Scholar

[16]

A. Marshall, "Principles of Economics,", 1st ed., (1890).   Google Scholar

[17]

M. Miller and M. Salmon, Dynamic games and the time inconsistency of optimal policy in open economics,, The Economic Journal, 95 (1985), 124.  doi: 10.2307/2232876.  Google Scholar

[18]

I. Palacios-Huerta, Time-inconsistent preferences in Adam Smith and Davis Hume,, History of Political Economy, 35 (2003), 241.  doi: 10.1215/00182702-35-2-241.  Google Scholar

[19]

V. Pareto, "Manuel d'économie Politique,", Girard and Brieve, (1909).   Google Scholar

[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.  doi: 10.2307/2296458.  Google Scholar

[21]

R. A. Pollak, Consistent planning,, Review of Economic Studies, 35 (1968), 185.  doi: 10.2307/2296548.  Google Scholar

[22]

A. Smith, "The Theory of Moral Sentiments,", First Edition, (1759).   Google Scholar

[23]

R. H. Strotz, Myopia and inconsistency in dynamic utility maximization,, Review of Econ. Studies, 23 (1955), 165.  doi: 10.2307/2295722.  Google Scholar

[24]

L. Tesfatsion, Time inconsistency of benevolent government economics,, J. Public Economics, 31 (1986), 25.  doi: 10.1016/0047-2727(86)90070-8.  Google Scholar

[25]

J. Yong, A deterministic time-inconsistent optimal control problem -- An essentially cooperative approach,, Acta Appl. Math. Sinica, ().   Google Scholar

[26]

J. Yong, and X. Y. Zhou, "Stochastic Controls: Hamiltonian Systems and HJB Equations,", Applications of Mathematics (New York), (1999).   Google Scholar

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