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 & Related Fields, 2012, 2 (3) : 271-329. doi: 10.3934/mcrf.2012.2.271
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show all references

References:
[1]

Rev. Finan. Stud., 23 (2010), 2970-3016. doi: 10.1093/rfs/hhq028.  Google Scholar

[2]

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

[3]

T. Björk, A. Murgoci and X. Y. Zhou, Mean varaiance portfolio optimization with state dependent risk aversion,, Math. Finance, ().   Google Scholar

[4]

Books for Libraries Press, Freeport, New York, 1891. Google Scholar

[5]

J. Econ. Theory, 131 (2006), 134-156. doi: 10.1016/j.jet.2005.05.006.  Google Scholar

[6]

Math. Finan. Econ., 4 (2010), 29-55.  Google Scholar

[7]

Math. Finan. Econ., 2 (2008), 57-86.  Google Scholar

[8]

SIAM J. Financial Math., 3 (2012), 1-32. Google Scholar

[9]

2nd edition, Stochastic Modelling and Applied Probability, 25, Springer, New York, 2006.  Google Scholar

[10]

Prentice Hall, Inc., Englewood Cliffs, NJ, 1964.  Google Scholar

[11]

Review of Economic Studies, 47 (1980), 533-537. Google Scholar

[12]

J. Finan. Econ., 84 (2007), 2-39. Google Scholar

[13]

Econ. Theory, 29 (2006), 591-619. doi: 10.1007/s00199-005-0020-3.  Google Scholar

[14]

arXiv:1111.0818, 2011. Google Scholar

[15]

First edition, 1739; Reprint, Oxford Univ. Press, New York, 1978. Google Scholar

[16]

McMillan, London, 1871. Google Scholar

[17]

J. Econ. Theory, 112 (2003), 353-364. doi: 10.1016/S0022-0531(03)00067-X.  Google Scholar

[18]

Quarterly J. Econ., 112 (1997), 443-477. doi: 10.1162/003355397555253.  Google Scholar

[19]

Probab. Theory Related Fields, 98 (1994), 339-359. doi: 10.1007/BF01192258.  Google Scholar

[20]

Lecture Notes in Math., 1702, Springer-Verlag, Berlin, 1999.  Google Scholar

[21]

in "The Works of Thomas Robert Malthus," Vols. 2-3 (eds. E. A. Wrigley and D. Souden), W. Pickering, London, 1986. Google Scholar

[22]

1st ed., 1890; 8th ed., Macmillan, London, 1920. Google Scholar

[23]

European J. Operational Research, 201 (2010), 860-872. doi: 10.1016/j.ejor.2009.04.005.  Google Scholar

[24]

Automatica J. IFAC, 47 (2011), 2626-2638. doi: 10.1016/j.automatica.2011.09.010.  Google Scholar

[25]

The Economic Journal, 95 (1985), 124-137. doi: 10.2307/2232876.  Google Scholar

[26]

History of Political Economy, 35 (2003), 241-268. doi: 10.1215/00182702-35-2-241.  Google Scholar

[27]

Girard and Brieve, Paris, 1909. Google Scholar

[28]

Review of Economic Studies, 40 (1973), 391-401. Google Scholar

[29]

Review of Economic Studies, 35 (1968), 185-199. doi: 10.2307/2296547.  Google Scholar

[30]

First edition, 1759; Reprint, Oxford Univ. Press, 1976. Google Scholar

[31]

Review of Econ. Studies, 23 (1955), 165-180. doi: 10.2307/2295722.  Google Scholar

[32]

J. Public Economics, 31 (1986), 25-52. doi: 10.1016/0047-2727(86)90070-8.  Google Scholar

[33]

Stoch. Proc. Appl., 116 (2006), 779-795. doi: 10.1016/j.spa.2006.01.005.  Google Scholar

[34]

Prob. Theory Rel. Fields, 142 (2008), 21-77. doi: 10.1007/s00440-007-0098-6.  Google Scholar

[35]

Math. Control Related Fields, 1 (2011), 83-118.  Google Scholar

[36]

Acta Math. Appl. Sinica Engl. Ser., 28 (2012), 1-30.  Google Scholar

[37]

Applications of Mathematics (New York), 43, Springer-Verlag, New York, 1999.  Google Scholar

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