September  2015, 5(3): 651-678. doi: 10.3934/mcrf.2015.5.651

Time-inconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation

1. 

School of Mathematics, Shandong University, Jinan 250100, China

Received  February 2014 Revised  July 2014 Published  July 2015

In this paper, we study a class of time-inconsistent optimal control problems with random coefficients. By the method of multi-person differential games, a family of parameterized backward stochastic partial differential equations, called the stochastic equilibrium Hamilton-Jacobi-Bellman equation, is derived for the equilibrium value function of this problem. Under appropriate conditions, we obtain the wellposedness of such an equation and construct the time-consistent equilibrium strategy of closed-loop. Besides, we investigate the linear-quadratic problem as a special and important case.
Citation: Haiyang Wang, Zhen Wu. Time-inconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation. Mathematical Control and Related Fields, 2015, 5 (3) : 651-678. doi: 10.3934/mcrf.2015.5.651
References:
[1]

T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochastic control problem, work in progress.

[2]

T. Björk, A. Murgoci and X. Y. Zhou, Mean variance portfolio optimization with state dependent risk aversion, Math. Finance, 24 (2014), 1-24. doi: 10.1111/j.1467-9965.2011.00515.x.

[3]

I. Ekeland and T. Pirvu, Investment and consumption without commitment, Math. Finan. Econ., 2 (2008), 57-86. doi: 10.1007/s11579-008-0014-6.

[4]

I. Ekeland and A. Lazrak, The golden rule when preferences are time inconsistent, Math. Finan. Econ., 4 (2010), 29-55. doi: 10.1007/s11579-010-0034-x.

[5]

I. Ekeland, O. Mbodji and T. Pirvu, Time-consistent portfolio management, SIAM J. Financial Math., 3 (2012), 1-32. doi: 10.1137/100810034.

[6]

S. Goldman, Consistent plans, Review of Economic Studies, 47 (1980), 533-537. doi: 10.2307/2297304.

[7]

Y. Hu, H. Q. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control, SIAM J. Control Optim., 50 (2012), 1548-1572. doi: 10.1137/110853960.

[8]

J. Ma and J. M. Yong, On linear, degenerate backward stochastic partial differential equations, Probab. Theory Related Fields, 113 (1999), 135-170. doi: 10.1007/s004400050205.

[9]

J. Ma, H. Yin and J. F. Zhang, On non-Markovian forward-backward SDEs and backward stochastic PDEs, Stochastic Processes and their Applications, 122 (2012), 3980-4004. doi: 10.1016/j.spa.2012.08.002.

[10]

J. Ma, Z. Wu, D. T. Zhang and J. F. Zhang, On wellposedness of forward-backward SDEs-a unified approach, Ann. Appl. Probab., 25(2015), 2168-2214.

[11]

I. Palacios-Huerta, Time-inconsistent preferences in Adam Smith and David Hume, History of Political Economy, 35 (2003), 241-268. doi: 10.1215/00182702-35-2-241.

[12]

E. Pardoux, Equations Aux Derivées Partielles Stochastiques Non Linéaires Monotones, Thèse d'Etat a l'Université Paris Sud, Paris, FR, 1975.

[13]

B. Peleg and M. Yaari, On the existence of a consistent course of action when tastes are changing, Review of Economic Studies, 40 (1973), 391-401. doi: 10.2307/2296458.

[14]

S. G. Peng, Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control and Optimization, 30 (1992), 284-304. doi: 10.1137/0330018.

[15]

R. Pollak, Consistent planning, Rev. Econ. Stud., 35 (1968), 201-208. doi: 10.2307/2296548.

[16]

R. Strotz, Myopia and inconsistency in dynamic utility maximization, Rev. Econ. Stud., 23 (1955), 165-180. doi: 10.2307/2295722.

[17]

J. M. Yong, A deterministic linear quadratic time-inconsistent optimal control problem, Math. Control Related Fields, 1 (2011), 83-118. doi: 10.3934/mcrf.2011.1.83.

[18]

J. M. Yong, Deterministic time-inconsistent optimal control problems-An essentially cooperative approach, Acta Math. Appl. Sinica Engl. Ser., 28 (2012), 1-30. doi: 10.1007/s10255-012-0120-3.

[19]

J. M. Yong, Time-inconsistent optimal control problems and the Equilibrium HJB equation, Mathematical Control and Related Fields, 2 (2012), 271-329. doi: 10.3934/mcrf.2012.2.271.

show all references

References:
[1]

T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochastic control problem, work in progress.

[2]

T. Björk, A. Murgoci and X. Y. Zhou, Mean variance portfolio optimization with state dependent risk aversion, Math. Finance, 24 (2014), 1-24. doi: 10.1111/j.1467-9965.2011.00515.x.

[3]

I. Ekeland and T. Pirvu, Investment and consumption without commitment, Math. Finan. Econ., 2 (2008), 57-86. doi: 10.1007/s11579-008-0014-6.

[4]

I. Ekeland and A. Lazrak, The golden rule when preferences are time inconsistent, Math. Finan. Econ., 4 (2010), 29-55. doi: 10.1007/s11579-010-0034-x.

[5]

I. Ekeland, O. Mbodji and T. Pirvu, Time-consistent portfolio management, SIAM J. Financial Math., 3 (2012), 1-32. doi: 10.1137/100810034.

[6]

S. Goldman, Consistent plans, Review of Economic Studies, 47 (1980), 533-537. doi: 10.2307/2297304.

[7]

Y. Hu, H. Q. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control, SIAM J. Control Optim., 50 (2012), 1548-1572. doi: 10.1137/110853960.

[8]

J. Ma and J. M. Yong, On linear, degenerate backward stochastic partial differential equations, Probab. Theory Related Fields, 113 (1999), 135-170. doi: 10.1007/s004400050205.

[9]

J. Ma, H. Yin and J. F. Zhang, On non-Markovian forward-backward SDEs and backward stochastic PDEs, Stochastic Processes and their Applications, 122 (2012), 3980-4004. doi: 10.1016/j.spa.2012.08.002.

[10]

J. Ma, Z. Wu, D. T. Zhang and J. F. Zhang, On wellposedness of forward-backward SDEs-a unified approach, Ann. Appl. Probab., 25(2015), 2168-2214.

[11]

I. Palacios-Huerta, Time-inconsistent preferences in Adam Smith and David Hume, History of Political Economy, 35 (2003), 241-268. doi: 10.1215/00182702-35-2-241.

[12]

E. Pardoux, Equations Aux Derivées Partielles Stochastiques Non Linéaires Monotones, Thèse d'Etat a l'Université Paris Sud, Paris, FR, 1975.

[13]

B. Peleg and M. Yaari, On the existence of a consistent course of action when tastes are changing, Review of Economic Studies, 40 (1973), 391-401. doi: 10.2307/2296458.

[14]

S. G. Peng, Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control and Optimization, 30 (1992), 284-304. doi: 10.1137/0330018.

[15]

R. Pollak, Consistent planning, Rev. Econ. Stud., 35 (1968), 201-208. doi: 10.2307/2296548.

[16]

R. Strotz, Myopia and inconsistency in dynamic utility maximization, Rev. Econ. Stud., 23 (1955), 165-180. doi: 10.2307/2295722.

[17]

J. M. Yong, A deterministic linear quadratic time-inconsistent optimal control problem, Math. Control Related Fields, 1 (2011), 83-118. doi: 10.3934/mcrf.2011.1.83.

[18]

J. M. Yong, Deterministic time-inconsistent optimal control problems-An essentially cooperative approach, Acta Math. Appl. Sinica Engl. Ser., 28 (2012), 1-30. doi: 10.1007/s10255-012-0120-3.

[19]

J. M. Yong, Time-inconsistent optimal control problems and the Equilibrium HJB equation, Mathematical Control and Related Fields, 2 (2012), 271-329. doi: 10.3934/mcrf.2012.2.271.

[1]

Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435

[2]

Libin Mou, Jiongmin Yong. Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method. Journal of Industrial and Management Optimization, 2006, 2 (1) : 95-117. doi: 10.3934/jimo.2006.2.95

[3]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control and Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97

[4]

Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001

[5]

Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control and Related Fields, 2022, 12 (2) : 371-404. doi: 10.3934/mcrf.2021026

[6]

Tyrone E. Duncan. Some partially observed multi-agent linear exponential quadratic stochastic differential games. Evolution Equations and Control Theory, 2018, 7 (4) : 587-597. doi: 10.3934/eect.2018028

[7]

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

[8]

Yu Li, Kok Lay Teo, Shuhua Zhang. A new feedback form of open-loop Stackelberg strategy in a general linear-quadratic differential game. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022105

[9]

Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040

[10]

Rui Mu, Zhen Wu. Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs. Mathematical Control and Related Fields, 2017, 7 (2) : 289-304. doi: 10.3934/mcrf.2017010

[11]

Tianxiao Wang. Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I. Mathematical Control and Related Fields, 2019, 9 (2) : 385-409. doi: 10.3934/mcrf.2019018

[12]

Weijun Meng, Jingtao Shi. A linear quadratic stochastic Stackelberg differential game with time delay. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021035

[13]

Shimin Li, Jaume Llibre. On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5885-5901. doi: 10.3934/dcdsb.2019111

[14]

Alain Bensoussan, Shaokuan Chen, Suresh P. Sethi. Linear quadratic differential games with mixed leadership: The open-loop solution. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 95-108. doi: 10.3934/naco.2013.3.95

[15]

Shigeaki Koike, Hiroaki Morimoto, Shigeru Sakaguchi. A linear-quadratic control problem with discretionary stopping. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 261-277. doi: 10.3934/dcdsb.2007.8.261

[16]

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053

[17]

Martino Bardi. Explicit solutions of some linear-quadratic mean field games. Networks and Heterogeneous Media, 2012, 7 (2) : 243-261. doi: 10.3934/nhm.2012.7.243

[18]

Xingwu Chen, Jaume Llibre, Weinian Zhang. Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3953-3965. doi: 10.3934/dcdsb.2017203

[19]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[20]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515

2021 Impact Factor: 1.141

Metrics

  • PDF downloads (292)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]