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A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion
Department of Mathematics, University of Bordj Bou Arreridj, 34000, Algeria |
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.
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G. Ainslie, Specious reward: A behavioral theory of impulsiveness and impulse control, Psychological Bulletin, 82 (1975), 463-496. Google Scholar |
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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 |
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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. |
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R. J. Barro, Ramsey meets Laibson in the neoclassical growth model, Quarterly Journal of Economics, 114 (1999), 1125-1152. Google Scholar |
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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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
S. M. Goldman,
Consistent plans, Review of Financial Studies, 47 (1980), 533-537.
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[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[29] |
R. A. Pollak,
Consistent planning, Review of Economic Studies, 35 (1968), 201-208.
doi: 10.2307/2296548. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
S. M. Goldman,
Consistent plans, Review of Financial Studies, 47 (1980), 533-537.
doi: 10.2307/2297304. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[29] |
R. A. Pollak,
Consistent planning, Review of Economic Studies, 35 (1968), 201-208.
doi: 10.2307/2296548. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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