# American Institute of Mathematical Sciences

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

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

 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.

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] Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control & Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022 [2] Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control & Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021 [3] Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499 [4] Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control & Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018 [5] Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195 [6] Jiaqin Wei, Danping Li, Yan Zeng. Robust optimal consumption-investment strategy with non-exponential discounting. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-24. doi: 10.3934/jimo.2018147 [7] Dejian Chang, Zhen Wu. Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance. Journal of Industrial & Management Optimization, 2015, 11 (1) : 27-40. doi: 10.3934/jimo.2015.11.27 [8] Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial & Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067 [9] Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581 [10] H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77 [11] John A. D. Appleby, Alexandra Rodkina, Henri Schurz. Pathwise non-exponential decay rates of solutions of scalar nonlinear stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 667-696. doi: 10.3934/dcdsb.2006.6.667 [12] Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174 [13] Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161 [14] 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 [15] H. O. Fattorini. The maximum principle in infinite dimension. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557 [16] Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401 [17] Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221 [18] Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335 [19] Andrzej Nowakowski, Jan Sokolowski. On dual dynamic programming in shape control. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2473-2485. doi: 10.3934/cpaa.2012.11.2473 [20] Thomas Leroy. Relativistic transfer equations: Comparison principle and convergence to the non-equilibrium regime. Kinetic & Related Models, 2015, 8 (4) : 725-763. doi: 10.3934/krm.2015.8.725

2018 Impact Factor: 1.292

## Metrics

• PDF downloads (42)
• HTML views (319)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]