Linear-quadratic optimal control problems for mean-field stochastic differential equations — time-consistent solutions
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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.References
- S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Rev. Finan. Stud. 23 (2010), 2970–3016.
- T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochasitc control problem, working paper.
- 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, DOI 10.1111/j.1467-9965.2011.00515.x
- V. S. Borkar and K. Suresh Kumar, McKean-Vlasov limit in portfolio optimization, Stoch. Anal. Appl. 28 (2010), no. 5, 884–906. MR 2739322, DOI 10.1080/07362994.2010.482836
- 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, DOI 10.1016/j.spa.2009.05.002
- Andrew Caplin and John Leahy, The recursive approach to time inconsistency, J. Econom. Theory 131 (2006), no. 1, 134–156. MR 2267047, DOI 10.1016/j.jet.2005.05.006
- 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, DOI 10.1007/s11579-012-0089-y
- Ivar Ekeland and Ali Lazrak, The golden rule when preferences are time inconsistent, Math. Financ. Econ. 4 (2010), no. 1, 29–55. MR 2746576, DOI 10.1007/s11579-010-0034-x
- Ivar Ekeland and Traian A. Pirvu, Investment and consumption without commitment, Math. Financ. Econ. 2 (2008), no. 1, 57–86. MR 2461340, DOI 10.1007/s11579-008-0014-6
- Ivar Ekeland, Oumar Mbodji, and Traian A. Pirvu, Time-consistent portfolio management, SIAM J. Financial Math. 3 (2012), no. 1, 1–32. MR 2968026, DOI 10.1137/100810034
- S. M. Goldman, Consistent plans, Review of Economic Studies 47 (1980), 533–537.
- S. R. Grenadier and N. Wang, Investment under uncertainty and time-inconsistent preferences, J. Finan. Econ. 84 (2007), 2–39.
- 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, DOI 10.1007/s00199-005-0020-3
- 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, DOI 10.1137/110853960
- Larry Karp and In Ho Lee, Time-consistent policies, J. Econom. Theory 112 (2003), no. 2, 353–364. MR 2008920, DOI 10.1016/S0022-0531(03)00067-X
- Peter E. Kloeden and Thomas Lorenz, Stochastic differential equations with nonlocal sample dependence, Stoch. Anal. Appl. 28 (2010), no. 6, 937–945. MR 2739325, DOI 10.1080/07362994.2010.515194
- D. Laibson, Golden eggs and hyperbolic discounting, Quarterly J. Econ. 112 (1997), 443–477.
- 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
- Jesús Marín-Solano and Jorge Navas, Consumption and portfolio rules for time-inconsistent investors, European J. Oper. Res. 201 (2010), no. 3, 860–872. MR 2552504, DOI 10.1016/j.ejor.2009.04.005
- Jesús Marín-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, DOI 10.1016/j.automatica.2011.09.010
- M. Miller and M. Salmon, Dynamic games and the time inconsistency of optimal policy in open economics, The Economic Journal 95 (1985), 124–137.
- R. A. Pollak, Consistent planning, Review of Economic Studies 35 (1968), 185–199.
- R. H. Strotz, Myopia and inconsistency in dynamic utility maximization, Review of Econ. Studies 23 (1955), 165–180.
- L. Tesfatsion, Time inconsistency of benevolent government economics, J. Public Economics 31 (1986), 25–52.
- Jiongmin Yong, A deterministic linear quadratic time-inconsistent optimal control problem, Math. Control Relat. Fields 1 (2011), no. 1, 83–118. MR 2822686, DOI 10.3934/mcrf.2011.1.83
- 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, DOI 10.1007/s10255-012-0120-3
- Jiongmin Yong, Time-inconsistent optimal control problems and the equilibrium HJB equation, Math. Control Relat. Fields 2 (2012), no. 3, 271–329. MR 2991570, DOI 10.3934/mcrf.2012.2.271
- 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, DOI 10.1137/120892477
- Jiongmin Yong and Xun Yu Zhou, Stochastic controls, Applications of Mathematics (New York), vol. 43, Springer-Verlag, New York, 1999. Hamiltonian systems and HJB equations. MR 1696772, DOI 10.1007/978-1-4612-1466-3
Additional Information
- Jiongmin Yong
- Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
- MR Author ID: 232631
- Email: jiongmin.yong@ucf.edu
- 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.
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 5467-5523
- MSC (2010): Primary 93E20, 49N10; Secondary 49N70
- DOI: https://doi.org/10.1090/tran/6502
- MathSciNet review: 3646768