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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

Jiongmin Yong 1,

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.
Keywords: linear-quadratic optimal control problem, Time-inconsistency, multi-person hierarchical differential games, time-consistent equilibrium strategy, Riccati--Volterra integral equation system..
Mathematics Subject Classification: Primary: 49L20, 49N10, 49N70; Secondary: 91A23, 91A6.
Citation: Jiongmin Yong. A deterministic linear quadratic time-inconsistent optimal control problem. Mathematical Control and Related Fields, 2011, 1 (1) : 83-118. doi: 10.3934/mcrf.2011.1.83
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, (). 

[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.

show all references

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, (). 

[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.

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