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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 and 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. 

[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.

[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. 

[5]

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

[6]

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

[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.

[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.

[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. 

[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).

[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. 

[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.

[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. 

[5]

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

[6]

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

[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.

[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.

[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. 

[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).

[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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