# American Institute of Mathematical Sciences

September  2012, 2(3): 271-329. doi: 10.3934/mcrf.2012.2.271

## Time-inconsistent optimal control problems and the equilibrium HJB equation

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

Received  April 2012 Revised  May 2012 Published  August 2012

A general time-inconsistent optimal control problem is considered for stochastic differential equations with deterministic coefficients. Under suitable conditions, a Hamilton-Jacobi-Bellman type equation is derived for the equilibrium value function of the problem. Well-posedness such an equation is studied, and time-consistent equilibrium strategies are constructed. As special cases, the linear-quadratic problem and a generalized Merton's portfolio problem are investigated.
Citation: Jiongmin Yong. Time-inconsistent optimal control problems and the equilibrium HJB equation. Mathematical Control and Related Fields, 2012, 2 (3) : 271-329. doi: 10.3934/mcrf.2012.2.271
##### References:
 [1] S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Rev. Finan. Stud., 23 (2010), 2970-3016. doi: 10.1093/rfs/hhq028. [2] T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochasitic control problem,, working paper., (). [3] T. Björk, A. Murgoci and X. Y. Zhou, Mean varaiance portfolio optimization with state dependent risk aversion,, Math. Finance, (). [4] E. V. Böhm-Bawerk, "The Positive Theory of Capital,'' Books for Libraries Press, Freeport, New York, 1891. [5] A. Caplin and J. Leahy, The recursive approach to time inconsistency, J. Econ. Theory, 131 (2006), 134-156. doi: 10.1016/j.jet.2005.05.006. [6] I. Ekeland and A. Lazrak, The golden rule when preferences are time inconsistent, Math. Finan. Econ., 4 (2010), 29-55. [7] I. Ekeland and T. A. Pirvu, Investment and consumption without commitment, Math. Finan. Econ., 2 (2008), 57-86. [8] I. Ekeland, O. Mbodji and T. A. Pirvu, Time-consistent portfolio management, SIAM J. Financial Math., 3 (2012), 1-32. [9] W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,'' 2nd edition, Stochastic Modelling and Applied Probability, 25, Springer, New York, 2006. [10] A. Friedman, "Partial Differential Equations of Parabolic Type,'' Prentice Hall, Inc., Englewood Cliffs, NJ, 1964. [11] S. M. Goldman, Consistent plans, Review of Economic Studies, 47 (1980), 533-537. [12] S. R. Grenadier and N. Wang, Investment under uncertainty and time-inconsistent preferences, J. Finan. Econ., 84 (2007), 2-39. [13] P. J.-J. Herings and K. I. M. Rohde, Time-inconsistent preferences in a general equilibrium model, Econ. Theory, 29 (2006), 591-619. doi: 10.1007/s00199-005-0020-3. [14] Y. Hu, H. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control, arXiv:1111.0818, 2011. [15] D. Hume, "A Treatise of Human Nature,'' First edition, 1739; Reprint, Oxford Univ. Press, New York, 1978. [16] W. S. Jevons, "Theory of Political Economy,'' McMillan, London, 1871. [17] L. Karp and I. H. Lee, Time-consistent policies, J. Econ. Theory, 112 (2003), 353-364. doi: 10.1016/S0022-0531(03)00067-X. [18] D. Laibson, Golden eggs and hyperbolic discounting, Quarterly J. Econ., 112 (1997), 443-477. doi: 10.1162/003355397555253. [19] J. Ma, P. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitly-a four-step scheme, Probab. Theory Related Fields, 98 (1994), 339-359. doi: 10.1007/BF01192258. [20] J. Ma and J. Yong, "Forward-Backward Stochastic Differential Equations and Their Applications,'' Lecture Notes in Math., 1702, Springer-Verlag, Berlin, 1999. [21] A. Malthus, An essay on the principle of population, in "The Works of Thomas Robert Malthus," Vols. 2-3 (eds. E. A. Wrigley and D. Souden), W. Pickering, London, 1986. [22] A. Marshall, "Principles of Economics,'' 1st ed., 1890; 8th ed., Macmillan, London, 1920. [23] J. Marin-Solano and J. Navas, Consumption and portfolio rules for time-inconsistent investors, European J. Operational Research, 201 (2010), 860-872. doi: 10.1016/j.ejor.2009.04.005. [24] J. Marin-Solano and E. V. Shevkoplyas, Non-constant discounting and differential games with random time horizon, Automatica J. IFAC, 47 (2011), 2626-2638. doi: 10.1016/j.automatica.2011.09.010. [25] 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. [26] 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. [27] V. Pareto, "Manuel d'Économie Politique,'' Girard and Brieve, Paris, 1909. [28] 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. [29] R. A. Pollak, Consistent planning, Review of Economic Studies, 35 (1968), 185-199. doi: 10.2307/2296547. [30] A. Smith, "The Theory of Moral Sentiments,'' First edition, 1759; Reprint, Oxford Univ. Press, 1976. [31] R. H. Strotz, Myopia and inconsistency indynamic utility maximization, Review of Econ. Studies, 23 (1955), 165-180. doi: 10.2307/2295722. [32] L. Tesfatsion, Time inconsistency of benevolent government economics, J. Public Economics, 31 (1986), 25-52. doi: 10.1016/0047-2727(86)90070-8. [33] J. Yong, Backward stochastic Volterra integral equations and some related problems, Stoch. Proc. Appl., 116 (2006), 779-795. doi: 10.1016/j.spa.2006.01.005. [34] J. Yong, Well-posedness and regularity of backward stochastic Volterra integral equations, Prob. Theory Rel. Fields, 142 (2008), 21-77. doi: 10.1007/s00440-007-0098-6. [35] J. Yong, A deterministic linear quadratic time-inconsistent optimal control problem, Math. Control Related Fields, 1 (2011), 83-118. [36] J. Yong, Deterministic time-inconsistent optimal control problems-An essentially cooperative approach, Acta Math. Appl. Sinica Engl. Ser., 28 (2012), 1-30. [37] 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, Rev. Finan. Stud., 23 (2010), 2970-3016. doi: 10.1093/rfs/hhq028. [2] T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochasitic control problem,, working paper., (). [3] T. Björk, A. Murgoci and X. Y. Zhou, Mean varaiance portfolio optimization with state dependent risk aversion,, Math. Finance, (). [4] E. V. Böhm-Bawerk, "The Positive Theory of Capital,'' Books for Libraries Press, Freeport, New York, 1891. [5] A. Caplin and J. Leahy, The recursive approach to time inconsistency, J. Econ. Theory, 131 (2006), 134-156. doi: 10.1016/j.jet.2005.05.006. [6] I. Ekeland and A. Lazrak, The golden rule when preferences are time inconsistent, Math. Finan. Econ., 4 (2010), 29-55. [7] I. Ekeland and T. A. Pirvu, Investment and consumption without commitment, Math. Finan. Econ., 2 (2008), 57-86. [8] I. Ekeland, O. Mbodji and T. A. Pirvu, Time-consistent portfolio management, SIAM J. Financial Math., 3 (2012), 1-32. [9] W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,'' 2nd edition, Stochastic Modelling and Applied Probability, 25, Springer, New York, 2006. [10] A. Friedman, "Partial Differential Equations of Parabolic Type,'' Prentice Hall, Inc., Englewood Cliffs, NJ, 1964. [11] S. M. Goldman, Consistent plans, Review of Economic Studies, 47 (1980), 533-537. [12] S. R. Grenadier and N. Wang, Investment under uncertainty and time-inconsistent preferences, J. Finan. Econ., 84 (2007), 2-39. [13] P. J.-J. Herings and K. I. M. Rohde, Time-inconsistent preferences in a general equilibrium model, Econ. Theory, 29 (2006), 591-619. doi: 10.1007/s00199-005-0020-3. [14] Y. Hu, H. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control, arXiv:1111.0818, 2011. [15] D. Hume, "A Treatise of Human Nature,'' First edition, 1739; Reprint, Oxford Univ. Press, New York, 1978. [16] W. S. Jevons, "Theory of Political Economy,'' McMillan, London, 1871. [17] L. Karp and I. H. Lee, Time-consistent policies, J. Econ. Theory, 112 (2003), 353-364. doi: 10.1016/S0022-0531(03)00067-X. [18] D. Laibson, Golden eggs and hyperbolic discounting, Quarterly J. Econ., 112 (1997), 443-477. doi: 10.1162/003355397555253. [19] J. Ma, P. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitly-a four-step scheme, Probab. Theory Related Fields, 98 (1994), 339-359. doi: 10.1007/BF01192258. [20] J. Ma and J. Yong, "Forward-Backward Stochastic Differential Equations and Their Applications,'' Lecture Notes in Math., 1702, Springer-Verlag, Berlin, 1999. [21] A. Malthus, An essay on the principle of population, in "The Works of Thomas Robert Malthus," Vols. 2-3 (eds. E. A. Wrigley and D. Souden), W. Pickering, London, 1986. [22] A. Marshall, "Principles of Economics,'' 1st ed., 1890; 8th ed., Macmillan, London, 1920. [23] J. Marin-Solano and J. Navas, Consumption and portfolio rules for time-inconsistent investors, European J. Operational Research, 201 (2010), 860-872. doi: 10.1016/j.ejor.2009.04.005. [24] J. Marin-Solano and E. V. Shevkoplyas, Non-constant discounting and differential games with random time horizon, Automatica J. IFAC, 47 (2011), 2626-2638. doi: 10.1016/j.automatica.2011.09.010. [25] 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. [26] 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. [27] V. Pareto, "Manuel d'Économie Politique,'' Girard and Brieve, Paris, 1909. [28] 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. [29] R. A. Pollak, Consistent planning, Review of Economic Studies, 35 (1968), 185-199. doi: 10.2307/2296547. [30] A. Smith, "The Theory of Moral Sentiments,'' First edition, 1759; Reprint, Oxford Univ. Press, 1976. [31] R. H. Strotz, Myopia and inconsistency indynamic utility maximization, Review of Econ. Studies, 23 (1955), 165-180. doi: 10.2307/2295722. [32] L. Tesfatsion, Time inconsistency of benevolent government economics, J. Public Economics, 31 (1986), 25-52. doi: 10.1016/0047-2727(86)90070-8. [33] J. Yong, Backward stochastic Volterra integral equations and some related problems, Stoch. Proc. Appl., 116 (2006), 779-795. doi: 10.1016/j.spa.2006.01.005. [34] J. Yong, Well-posedness and regularity of backward stochastic Volterra integral equations, Prob. Theory Rel. Fields, 142 (2008), 21-77. doi: 10.1007/s00440-007-0098-6. [35] J. Yong, A deterministic linear quadratic time-inconsistent optimal control problem, Math. Control Related Fields, 1 (2011), 83-118. [36] J. Yong, Deterministic time-inconsistent optimal control problems-An essentially cooperative approach, Acta Math. Appl. Sinica Engl. Ser., 28 (2012), 1-30. [37] 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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