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Linear-quadratic optimal control problems for mean-field stochastic differential equations -- time-consistent solutions


Author: Jiongmin Yong
Journal: Trans. Amer. Math. Soc. 369 (2017), 5467-5523
MSC (2010): Primary 93E20, 49N10; Secondary 49N70
DOI: https://doi.org/10.1090/tran/6502
Published electronically: December 18, 2015
MathSciNet review: 3072755
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Abstract | References | Similar Articles | Additional Information

Abstract: Linear-quadratic optimal control problems are considered for
mean-field stochastic differential equations with deterministic coefficients.
Time-inconsistency feature of the problems is carefully investigated. Both open-loop and closed-loop equilibrium solutions are presented for such kinds of problems. Open-loop solutions are presented by means of the variational method with decoupling of forward-backward stochastic differential equations, which leads to a Riccati equation system lack of symmetry. Closed-loop solutions are presented by means of multi-person differential games, the limit of which leads to a Riccati equation system with a symmetric structure.


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  • [1] S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Rev. Finan. Stud. 23 (2010), 2970-3016.
  • [2] T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochasitc control problem, working paper.
  • [3] Tomas Björk, Agatha Murgoci, and Xun Yu Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Math. Finance 24 (2014), no. 1, 1-24. MR 3157686, https://doi.org/10.1111/j.1467-9965.2011.00515.x
  • [4] V. S. Borkar and K. Suresh Kumar, McKean-Vlasov limit in portfolio optimization, Stoch. Anal. Appl. 28 (2010), no. 5, 884-906. MR 2739322 (2011j:91245), https://doi.org/10.1080/07362994.2010.482836
  • [5] Rainer Buckdahn, Juan Li, and Shige Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Process. Appl. 119 (2009), no. 10, 3133-3154. MR 2568268 (2011d:60167), https://doi.org/10.1016/j.spa.2009.05.002
  • [6] Andrew Caplin and John Leahy, The recursive approach to time inconsistency, J. Econom. Theory 131 (2006), no. 1, 134-156. MR 2267047 (2007j:90034), https://doi.org/10.1016/j.jet.2005.05.006
  • [7] René Carmona, François Delarue, and Aimé Lachapelle, Control of McKean-Vlasov dynamics versus mean field games, Math. Financ. Econ. 7 (2013), no. 2, 131-166. MR 3045029, https://doi.org/10.1007/s11579-012-0089-y
  • [8] Ivar Ekeland and Ali Lazrak, The golden rule when preferences are time inconsistent, Math. Financ. Econ. 4 (2010), no. 1, 29-55. MR 2746576 (2012a:91137), https://doi.org/10.1007/s11579-010-0034-x
  • [9] Ivar Ekeland and Traian A. Pirvu, Investment and consumption without commitment, Math. Financ. Econ. 2 (2008), no. 1, 57-86. MR 2461340 (2010a:91131), https://doi.org/10.1007/s11579-008-0014-6
  • [10] Ivar Ekeland, Oumar Mbodji, and Traian A. Pirvu, Time-consistent portfolio management, SIAM J. Financial Math. 3 (2012), no. 1, 1-32. MR 2968026, https://doi.org/10.1137/100810034
  • [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. Jean-Jacques Herings and Kirsten I. M. Rohde, Time-inconsistent preferences in a general equilibrium model, Econom. Theory 29 (2006), no. 3, 591-619. MR 2272314 (2007m:91104), https://doi.org/10.1007/s00199-005-0020-3
  • [14] Ying Hu, Hanqing Jin, and Xun Yu Zhou, Time-inconsistent stochastic linear-quadratic control, SIAM J. Control Optim. 50 (2012), no. 3, 1548-1572. MR 2968066, https://doi.org/10.1137/110853960
  • [15] Larry Karp and In Ho Lee, Time-consistent policies, J. Econom. Theory 112 (2003), no. 2, 353-364. MR 2008920 (2004g:91110), https://doi.org/10.1016/S0022-0531(03)00067-X
  • [16] Peter E. Kloeden and Thomas Lorenz, Stochastic differential equations with nonlocal sample dependence, Stoch. Anal. Appl. 28 (2010), no. 6, 937-945. MR 2739325 (2012c:60174), https://doi.org/10.1080/07362994.2010.515194
  • [17] D. Laibson, Golden eggs and hyperbolic discounting, Quarterly J. Econ. 112 (1997), 443-477.
  • [18] Jin Ma and Jiongmin Yong, Forward-backward stochastic differential equations and their applications, Lecture Notes in Mathematics, vol. 1702, Springer-Verlag, Berlin, 1999. MR 1704232 (2000k:60118)
  • [19] Jesus Marin-Solano and Jorge Navas, Consumption and portfolio rules for time-inconsistent investors, European J. Oper. Res. 201 (2010), no. 3, 860-872. MR 2552504 (2011h:91192), https://doi.org/10.1016/j.ejor.2009.04.005
  • [20] Jesus Marin-Solano and Ekaterina V. Shevkoplyas, Non-constant discounting and differential games with random time horizon, Automatica J. IFAC 47 (2011), no. 12, 2626-2638. MR 2886931, https://doi.org/10.1016/j.automatica.2011.09.010
  • [21] M. Miller and M. Salmon, Dynamic games and the time inconsistency of optimal policy in open economics, The Economic Journal 95 (1985), 124-137.
  • [22] R. A. Pollak, Consistent planning, Review of Economic Studies 35 (1968), 185-199.
  • [23] R. H. Strotz, Myopia and inconsistency in dynamic utility maximization, Review of Econ. Studies 23 (1955), 165-180.
  • [24] L. Tesfatsion, Time inconsistency of benevolent government economics, J. Public Economics 31 (1986), 25-52.
  • [25] Jiongmin Yong, A deterministic linear quadratic time-inconsistent optimal control problem, Math. Control Relat. Fields 1 (2011), no. 1, 83-118. MR 2822686 (2012k:49081), https://doi.org/10.3934/mcrf.2011.1.83
  • [26] Jiong-min Yong, Deterministic time-inconsistent optimal control problems--an essentially cooperative approach, Acta Math. Appl. Sin. Engl. Ser. 28 (2012), no. 1, 1-30. MR 2864348 (2012j:49098), https://doi.org/10.1007/s10255-012-0120-3
  • [27] Jiongmin Yong, Time-inconsistent optimal control problems and the equilibrium HJB equation, Math. Control Relat. Fields 2 (2012), no. 3, 271-329. MR 2991570, https://doi.org/10.3934/mcrf.2012.2.271
  • [28] Jiongmin Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations, SIAM J. Control Optim. 51 (2013), no. 4, 2809-2838. MR 3072755, https://doi.org/10.1137/120892477
  • [29] Jiongmin Yong and Xun Yu Zhou, Stochastic controls: Hamiltonian systems and HJB equations, Applications of Mathematics (New York), vol. 43, Springer-Verlag, New York, 1999. MR 1696772 (2001d:93004)

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Additional Information

Jiongmin Yong
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email: jiongmin.yong@ucf.edu

DOI: https://doi.org/10.1090/tran/6502
Keywords: Mean-field stochastic differential equation, linear-quadratic optimal control, time-inconsistency, equilibrium solution, Riccati equation, Lyapunov equation, $N$-person differential games
Received by editor(s): May 6, 2013
Received by editor(s) in revised form: June 13, 2014, and August 28, 2015
Published electronically: December 18, 2015
Additional Notes: This work was supported in part by NSF Grant DMS-1406776.
Article copyright: © Copyright 2015 American Mathematical Society

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