American Institute of Mathematical Sciences

Advanced
September  2019, 9(3): 541-570. doi: 10.3934/mcrf.2019025

A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion

Ishak Alia

Department of Mathematics, University of Bordj Bou Arreridj, 34000, Algeria

Received  August 2017 Revised  November 2018 Published  April 2019

In this paper, we study general time-inconsistent stochastic control models which are driven by a stochastic differential equation with random jumps. Specifically, the time-inconsistency arises from the presence of a non-exponential discount function in the objective functional. We consider equilibrium, instead of optimal, solution within the class of open-loop controls. We prove an equivalence relationship between our time-inconsistent problem and a time-consistent problem such that the equilibrium controls for the time-consistent problem coincide with the equilibrium controls for the time-inconsistent problem. We establish two general results which characterize the open-loop equilibrium controls. As special cases, a generalized Merton's portfolio problem and a linear-quadratic problem are discussed.

Keywords: Time-inconsistency, stochastic control, non-exponential discounting, equilibrium control, dynamic programming principle, stochastic maximum principle, Poisson point process.
Mathematics Subject Classification: Primary: 93E20, 91B08, 91B70; Secondary: 62P20, 60H30, 60H10, 49N90, 93E25.
Citation: Ishak Alia. A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion. Mathematical Control & Related Fields, 2019, 9 (3) : 541-570. doi: 10.3934/mcrf.2019025
References:
[1]

G. Ainslie, Specious reward: A behavioral theory of impulsiveness and impulse control, Psychological Bulletin, 82 (1975), 463-496.   Google Scholar

[2]

I. Alia, F. Chighoub, N. Khelfallah and J. Vives, Time-consistent investment and consumption strategies under a general discount function, preprint, arXiv: 1705.10602. Google Scholar

[3]

N. Azevedo, D. Pinheiro and G. W. Weber, Dynamic programming for a Markov-switching jump–diffusion, Journal of Computational and Applied Mathematics, 267 (2014), 1-19.  doi: 10.1016/j.cam.2014.01.021.  Google Scholar

[4]

R. J. Barro, Ramsey meets Laibson in the neoclassical growth model, Quarterly Journal of Economics, 114 (1999), 1125-1152.   Google Scholar

[5]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Review of Financial Studies, 23 (2010), 2970-3016.   Google Scholar

[6]

T. Björk and A. Murgoci, A general theory of Markovian time-inconsistent stochastic control problems, 2010. Available from: https://ssrn.com/abstract=1694759. Google Scholar

[7]

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

[8]

T. Björk and A. Murgoci, A theory of Markovian time-inconsistent stochastic control in discrete time, Finance and Stochastics, 18 (2014), 545-592.  doi: 10.1007/s00780-014-0234-y.  Google Scholar

[9]

T. Bjork, M. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance and Stochastics, 21 (2017), 331-360.  doi: 10.1007/s00780-017-0327-5.  Google Scholar

[10]

C. Czichowsky, Time-consistent mean-variance porftolio selection in discrete and continuous time, Finance and Stochastics, 17 (2013), 227-271.  doi: 10.1007/s00780-012-0189-9.  Google Scholar

[11]

B. Djehiche and M. Huang, A characterization of sub-game perfect Nash equilibria for SDEs of mean field type, Dynamic Games and Applications, 6 (2016), 55-81.  doi: 10.1007/s13235-015-0140-8.  Google Scholar

[12]

Y. Dong and R. Sircar, Time-inconsistent portfolio investment problems, in Stochastic Analysis and Applications, Springer, 100 (2014), 239–281. doi: 10.1007/978-3-319-11292-3_9.  Google Scholar

[13]

I. Ekeland and A. Lazrak, Equilibrium policies when preferences are time-inconsistent, preprint, arXiv: 0808.3790v1. Google Scholar

[14]

I. Ekeland and T. A. Pirvu, Investment and consumption without commitment, Mathematics and Financial Economics, 2 (2008), 57-86.  doi: 10.1007/s11579-008-0014-6.  Google Scholar

[15]

I. Ekeland, O. Mbodji and T. A. Pirvu, Time-consistent portfolio management, SIAM Journal on Financial Mathematics, 3 (2012), 1-32.  doi: 10.1137/100810034.  Google Scholar

[16]

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

[17]

Y. Hu, H. Jin and X. Y. Zhou, Time-inconsistent stochastic linear quadratic control, SIAM Journal on Control and Optimization, 50 (2012), 1548-1572.  doi: 10.1137/110853960.  Google Scholar

[18]

Y. Hu, H. Jin and X. Y. Zhou, Time-inconsistent stochastic linear quadratic control: Characterization and uniqueness of equilibrium, SIAM J. Control Optim., 55 (2017), 1261–1279, arXiv: 1504.01152. doi: 10.1137/15M1019040.  Google Scholar

[19]

Y. Hu, J. Huang and X. Li, Equilibrium for time-inconsistent stochastic linear–quadratic control under constraint, preprint, arXiv: 1703.09415v1. Google Scholar

[20]

P. Krusell and A. Smith, Consumption and savings decisions with quasi-geometric discounting, Econometrica, 71 (2003), 365-375.  doi: 10.1111/1468-0262.00400.  Google Scholar

[21]

F. E. Kydland and E. Prescott, Rules rather than discretion: The inconsistency of optimal plans, Journal of Political Economy, 85 (1997), 473-492.   Google Scholar

[22]

G. Loewenstein and D. Prelec, Anomalies in intertemporal choice: Evidence and an interpretation, Choices, Values, and Frames, (2019), 578–596. doi: 10.1017/CBO9780511803475.034.  Google Scholar

[23]

J. Marin-Solano and J. Navas, Consumption and portfolio rules for time-inconsistent investors, European Journal of Operational Research, 201 (2010), 860-872.  doi: 10.1016/j.ejor.2009.04.005.  Google Scholar

[24]

J. Marin-Solano and E. V. Shevkoplyas, Non-constant discounting and differential games with random time horizon, Automatica IFAC Journal, 47 (2011), 2626-2638.  doi: 10.1016/j.automatica.2011.09.010.  Google Scholar

[25]

Q. Meng, General linear quadratic optimal stochastic control problem driven by a Brownian motion and a Poisson random martingale measure with random coefficients, Stochastic Analysis and Applications, 32 (2014), 88-109.  doi: 10.1080/07362994.2013.845106.  Google Scholar

[26]

R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X.  Google Scholar

[27]

B. Oksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, 2$^{nd}$ edition, Springer, 2007. doi: 10.1007/978-3-540-69826-5.  Google Scholar

[28]

E. S. Phelps and R. A. Pollak, On second-best national saving and game-equilibrium growth, Studies in Macroeconomic Theory, (1980), 201–215. doi: 10.1016/B978-0-12-554002-5.50020-0.  Google Scholar

[29]

R. A. Pollak, Consistent planning, Review of Economic Studies, 35 (1968), 201-208.  doi: 10.2307/2296548.  Google Scholar

[30]

Y. Shen and T. K. Siu, The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem, Nonlinear Analysis, 86 (2013), 58-73.  doi: 10.1016/j.na.2013.02.029.  Google Scholar

[31]

R. Strotz, Myopia and inconsistency in dynamic utility maximization, Readings in Welfare Economics, (1973), 128–143. doi: 10.1007/978-1-349-15492-0_10.  Google Scholar

[32]

S. Tang and X. Li, Necessary conditions for optimal control for stochastic systems with random jumps, SIAM Journal on Control and Optimization, 32 (1994), 1447-1475.  doi: 10.1137/S0363012992233858.  Google Scholar

[33]

Q. Wei, J. Yong and Z. Yu, Time-inconsistent recursive stochastic optimal control problems, SIAM J. Control Optim., 55 (2017), 4156–4201, arXiv: 1606.03330v1. doi: 10.1137/16M1079415.  Google Scholar

[34]

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

[35]

J. Yong, Deterministic time-inconsistent optimal control problems-An essentially cooperative approach, Acta Mathematicae Applicatae Sinica (English Series), 28 (2012), 1-30.  doi: 10.1007/s10255-012-0120-3.  Google Scholar

[36]

J. 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.  Google Scholar

[37]

J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations–time-consistent solutions, Transactions of the American Mathematical, 369 (2017), 5467-5523.  doi: 10.1090/tran/6502.  Google Scholar

[38]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[39]

Q. Zhao, On Time-Inconsistent Investment and Dividend Problems, PhD thesis, Australia : Macquarie University, (2015). Google Scholar

[40]

Q. Zhao, Y. Shen and J. Wei, Consumption-investment strategies with non-exponential discounting and logarithmic utility, European Journal of Operational Research, 238 (2014), 824-835.  doi: 10.1016/j.ejor.2014.04.034.  Google Scholar

show all references

References:
[1]

G. Ainslie, Specious reward: A behavioral theory of impulsiveness and impulse control, Psychological Bulletin, 82 (1975), 463-496.   Google Scholar

[2]

I. Alia, F. Chighoub, N. Khelfallah and J. Vives, Time-consistent investment and consumption strategies under a general discount function, preprint, arXiv: 1705.10602. Google Scholar

[3]

N. Azevedo, D. Pinheiro and G. W. Weber, Dynamic programming for a Markov-switching jump–diffusion, Journal of Computational and Applied Mathematics, 267 (2014), 1-19.  doi: 10.1016/j.cam.2014.01.021.  Google Scholar

[4]

R. J. Barro, Ramsey meets Laibson in the neoclassical growth model, Quarterly Journal of Economics, 114 (1999), 1125-1152.   Google Scholar

[5]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Review of Financial Studies, 23 (2010), 2970-3016.   Google Scholar

[6]

T. Björk and A. Murgoci, A general theory of Markovian time-inconsistent stochastic control problems, 2010. Available from: https://ssrn.com/abstract=1694759. Google Scholar

[7]

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

[8]

T. Björk and A. Murgoci, A theory of Markovian time-inconsistent stochastic control in discrete time, Finance and Stochastics, 18 (2014), 545-592.  doi: 10.1007/s00780-014-0234-y.  Google Scholar

[9]

T. Bjork, M. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance and Stochastics, 21 (2017), 331-360.  doi: 10.1007/s00780-017-0327-5.  Google Scholar

[10]

C. Czichowsky, Time-consistent mean-variance porftolio selection in discrete and continuous time, Finance and Stochastics, 17 (2013), 227-271.  doi: 10.1007/s00780-012-0189-9.  Google Scholar

[11]

B. Djehiche and M. Huang, A characterization of sub-game perfect Nash equilibria for SDEs of mean field type, Dynamic Games and Applications, 6 (2016), 55-81.  doi: 10.1007/s13235-015-0140-8.  Google Scholar

[12]

Y. Dong and R. Sircar, Time-inconsistent portfolio investment problems, in Stochastic Analysis and Applications, Springer, 100 (2014), 239–281. doi: 10.1007/978-3-319-11292-3_9.  Google Scholar

[13]

I. Ekeland and A. Lazrak, Equilibrium policies when preferences are time-inconsistent, preprint, arXiv: 0808.3790v1. Google Scholar

[14]

I. Ekeland and T. A. Pirvu, Investment and consumption without commitment, Mathematics and Financial Economics, 2 (2008), 57-86.  doi: 10.1007/s11579-008-0014-6.  Google Scholar

[15]

I. Ekeland, O. Mbodji and T. A. Pirvu, Time-consistent portfolio management, SIAM Journal on Financial Mathematics, 3 (2012), 1-32.  doi: 10.1137/100810034.  Google Scholar

[16]

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

[17]

Y. Hu, H. Jin and X. Y. Zhou, Time-inconsistent stochastic linear quadratic control, SIAM Journal on Control and Optimization, 50 (2012), 1548-1572.  doi: 10.1137/110853960.  Google Scholar

[18]

Y. Hu, H. Jin and X. Y. Zhou, Time-inconsistent stochastic linear quadratic control: Characterization and uniqueness of equilibrium, SIAM J. Control Optim., 55 (2017), 1261–1279, arXiv: 1504.01152. doi: 10.1137/15M1019040.  Google Scholar

[19]

Y. Hu, J. Huang and X. Li, Equilibrium for time-inconsistent stochastic linear–quadratic control under constraint, preprint, arXiv: 1703.09415v1. Google Scholar

[20]

P. Krusell and A. Smith, Consumption and savings decisions with quasi-geometric discounting, Econometrica, 71 (2003), 365-375.  doi: 10.1111/1468-0262.00400.  Google Scholar

[21]

F. E. Kydland and E. Prescott, Rules rather than discretion: The inconsistency of optimal plans, Journal of Political Economy, 85 (1997), 473-492.   Google Scholar

[22]

G. Loewenstein and D. Prelec, Anomalies in intertemporal choice: Evidence and an interpretation, Choices, Values, and Frames, (2019), 578–596. doi: 10.1017/CBO9780511803475.034.  Google Scholar

[23]

J. Marin-Solano and J. Navas, Consumption and portfolio rules for time-inconsistent investors, European Journal of Operational Research, 201 (2010), 860-872.  doi: 10.1016/j.ejor.2009.04.005.  Google Scholar

[24]

J. Marin-Solano and E. V. Shevkoplyas, Non-constant discounting and differential games with random time horizon, Automatica IFAC Journal, 47 (2011), 2626-2638.  doi: 10.1016/j.automatica.2011.09.010.  Google Scholar

[25]

Q. Meng, General linear quadratic optimal stochastic control problem driven by a Brownian motion and a Poisson random martingale measure with random coefficients, Stochastic Analysis and Applications, 32 (2014), 88-109.  doi: 10.1080/07362994.2013.845106.  Google Scholar

[26]

R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X.  Google Scholar

[27]

B. Oksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, 2$^{nd}$ edition, Springer, 2007. doi: 10.1007/978-3-540-69826-5.  Google Scholar

[28]

E. S. Phelps and R. A. Pollak, On second-best national saving and game-equilibrium growth, Studies in Macroeconomic Theory, (1980), 201–215. doi: 10.1016/B978-0-12-554002-5.50020-0.  Google Scholar

[29]

R. A. Pollak, Consistent planning, Review of Economic Studies, 35 (1968), 201-208.  doi: 10.2307/2296548.  Google Scholar

[30]

Y. Shen and T. K. Siu, The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem, Nonlinear Analysis, 86 (2013), 58-73.  doi: 10.1016/j.na.2013.02.029.  Google Scholar

[31]

R. Strotz, Myopia and inconsistency in dynamic utility maximization, Readings in Welfare Economics, (1973), 128–143. doi: 10.1007/978-1-349-15492-0_10.  Google Scholar

[32]

S. Tang and X. Li, Necessary conditions for optimal control for stochastic systems with random jumps, SIAM Journal on Control and Optimization, 32 (1994), 1447-1475.  doi: 10.1137/S0363012992233858.  Google Scholar

[33]

Q. Wei, J. Yong and Z. Yu, Time-inconsistent recursive stochastic optimal control problems, SIAM J. Control Optim., 55 (2017), 4156–4201, arXiv: 1606.03330v1. doi: 10.1137/16M1079415.  Google Scholar

[34]

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

[35]

J. Yong, Deterministic time-inconsistent optimal control problems-An essentially cooperative approach, Acta Mathematicae Applicatae Sinica (English Series), 28 (2012), 1-30.  doi: 10.1007/s10255-012-0120-3.  Google Scholar

[36]

J. 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.  Google Scholar

[37]

J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations–time-consistent solutions, Transactions of the American Mathematical, 369 (2017), 5467-5523.  doi: 10.1090/tran/6502.  Google Scholar

[38]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[39]

Q. Zhao, On Time-Inconsistent Investment and Dividend Problems, PhD thesis, Australia : Macquarie University, (2015). Google Scholar

[40]

Q. Zhao, Y. Shen and J. Wei, Consumption-investment strategies with non-exponential discounting and logarithmic utility, European Journal of Operational Research, 238 (2014), 824-835.  doi: 10.1016/j.ejor.2014.04.034.  Google Scholar

[1]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[2]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008
[3]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[4]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[5]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023
[6]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[7]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329
[8]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183
[9]

Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313
[10]

Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208
[11]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088
[12]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301
[13]

A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909
[14]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175
[15]

Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021004
[16]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113
[17]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1
[18]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006
[19]

Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems & Imaging, 2012, 6 (4) : 565-598. doi: 10.3934/ipi.2012.6.565
[20]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021

2019 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (216)
  • HTML views (598)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences