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 & 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., (). Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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. doi: 10.1016/j.spa.2012.08.002. Google Scholar

[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. Google Scholar

[11]

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

[12]

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

[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. doi: 10.2307/2296458. Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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. doi: 10.1016/j.spa.2012.08.002. Google Scholar

[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. Google Scholar

[11]

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

[12]

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

[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. doi: 10.2307/2296458. Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

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