American Institute of Mathematical Sciences

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

Time-inconsistent optimal control problems and the equilibrium HJB equation

Jiongmin Yong 1,

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.
Keywords: Time-inconsistent optimal control problem, forward-backward stochastic differential equation., equilibrium value function, equilibrium Hamilton-Jacobi-Bellman equation.
Mathematics Subject Classification: Primary: 93E20, 49L20, 49N10, 49N70; Secondary: 35Q93,91A23, 91A6.
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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