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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 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.

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##### 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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