# American Institute of Mathematical Sciences

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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] S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation,, Preprint, (2008). [2] L. D. Berkovitz, "Optimal Control Theory,", Applied Mathematical Sciences, 12 (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, (1891). [5] I. Ekeland and A. Lazrak, Being serious about non-commitment: subgame perfect equilibrium in continuous time,, preprint, (2008). [6] I. Ekeland and T. Privu, Investment and consumption without commitment,, preprint, (2007). [7] S. M. Goldman, Consistent plans,, Review of Economic Studies, 47 (1980), 533. 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). [11] W. S. Jevons, "Theory of Political Economy,", Mcmillan, (1871). [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. [13] D. Laibson, Golden eggs and hyperbolic discounting,, Quarterly J. Econ., 112 (1997), 443. doi: 10.1162/003355397555253. [14] A. Malthus, An essay on the principle of population, 1826,, in, 2 (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. doi: 10.1016/j.jedc.2008.08.008. [16] A. Marshall, "Principles of Economics,", 1st ed., (1890). [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. [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. [19] V. Pareto, "Manuel d'économie Politique,", Girard and Brieve, (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. doi: 10.2307/2296458. [21] R. A. Pollak, Consistent planning,, Review of Economic Studies, 35 (1968), 185. doi: 10.2307/2296548. [22] A. Smith, "The Theory of Moral Sentiments,", First Edition, (1759). [23] R. H. Strotz, Myopia and inconsistency in dynamic utility maximization,, Review of Econ. Studies, 23 (1955), 165. doi: 10.2307/2295722. [24] L. Tesfatsion, Time inconsistency of benevolent government economics,, J. Public Economics, 31 (1986), 25. 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), (1999).

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##### References:
 [1] S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation,, Preprint, (2008). [2] L. D. Berkovitz, "Optimal Control Theory,", Applied Mathematical Sciences, 12 (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, (1891). [5] I. Ekeland and A. Lazrak, Being serious about non-commitment: subgame perfect equilibrium in continuous time,, preprint, (2008). [6] I. Ekeland and T. Privu, Investment and consumption without commitment,, preprint, (2007). [7] S. M. Goldman, Consistent plans,, Review of Economic Studies, 47 (1980), 533. 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). [11] W. S. Jevons, "Theory of Political Economy,", Mcmillan, (1871). [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. [13] D. Laibson, Golden eggs and hyperbolic discounting,, Quarterly J. Econ., 112 (1997), 443. doi: 10.1162/003355397555253. [14] A. Malthus, An essay on the principle of population, 1826,, in, 2 (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. doi: 10.1016/j.jedc.2008.08.008. [16] A. Marshall, "Principles of Economics,", 1st ed., (1890). [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. [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. [19] V. Pareto, "Manuel d'économie Politique,", Girard and Brieve, (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. doi: 10.2307/2296458. [21] R. A. Pollak, Consistent planning,, Review of Economic Studies, 35 (1968), 185. doi: 10.2307/2296548. [22] A. Smith, "The Theory of Moral Sentiments,", First Edition, (1759). [23] R. H. Strotz, Myopia and inconsistency in dynamic utility maximization,, Review of Econ. Studies, 23 (1955), 165. doi: 10.2307/2295722. [24] L. Tesfatsion, Time inconsistency of benevolent government economics,, J. Public Economics, 31 (1986), 25. 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), (1999).
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