doi: 10.3934/mcrf.2021053
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Open-loop equilibriums for a general class of time-inconsistent stochastic optimal control problems

Institute of Optics and Precision Mechanics, Ferhat Abbas University Setif 1, 19000 Setif, Algeria

Received  February 2021 Revised  August 2021 Early access October 2021

This paper studies open-loop equilibriums for a general class of time-inconsistent stochastic control problems under jump-diffusion SDEs with deterministic coefficients. Inspired by the idea of Four-Step-Scheme for forward-backward stochastic differential equations with jumps (FBSDEJs, for short), we derive two systems of integro-partial differential equations (IPDEs, for short). Then, we rigorously prove a verification theorem which provides a sufficient condition for open-loop equilibrium strategies. As special cases, a mean-variance portfolio selection problem and a time-inconsistent problem under non-exponential discounting are discussed.

Citation: Ishak Alia. Open-loop equilibriums for a general class of time-inconsistent stochastic optimal control problems. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021053
References:
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I. Alia, A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion, Math. Control Relat. Fields, 9 (2019), 541-570.  doi: 10.3934/mcrf.2019025.  Google Scholar

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I. Alia, Time-inconsistent stochastic optimal control problems: A backward stochastic partial differential equations approach, Math. Control Relat. Fields, 10 (2020), 785-826.  doi: 10.3934/mcrf.2020020.  Google Scholar

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T. Björk and A. Murgoci, A general theory of Markovian time-inconsistent stochastic control problems, \emphSSRN, (2010), 55pp, Available from: https://ssrn.com/abstract=1694759. Google Scholar

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

[6]

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

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S. Christensen and K. Lindensjö, On time-inconsistent stopping problems and mixed strategy stopping times, Stochastic Process. Appl., 130 (2020), 2886-2917.  doi: 10.1016/j.spa.2019.08.010.  Google Scholar

[8]

S. Christensen and K. Lindensjö, Time-inconsistent stopping, myopic adjustment & equilibrium stability: With a mean–variance application, Banach Center Publications, 122 (2020), 53-76.  doi: 10.4064/bc122-4.  Google Scholar

[9]

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

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

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F. Dou and Q. Lü, Time-inconsistent linear quadratic optimal control problems for stochastic evolution equations, SIAM J. Control Optim., 58 (2020), 485-509.  doi: 10.1137/19M1250339.  Google Scholar

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Y. Hamaguchi, Small-time solvability of a flow of forward-backward stochastic differential equations, Appl. Math. Optim., 84 (2021), 567-588.  doi: 10.1007/s00245-020-09654-7.  Google Scholar

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C. Hernández and D. Possamaï, Me, myself and I: A general theory of non–Markovian time–inconsistent stochastic control for sophisticated agents, preprint, arXiv: 2002.12572v2. Google Scholar

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

[24]

Y. Hu, J. Huang and X. Li, Equilibrium for time-inconsistent stochastic linear–quadratic control under constraint, preprint, arXiv: 1703.09415v1. doi: 10.1137/15M1019040.  Google Scholar

[25]

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

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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

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K. Lindensjö, A regular equilibrium solves the extended HJB system, Oper. Res. Lett., 47 (2019), 427-432.  doi: 10.1016/j.orl.2019.07.011.  Google Scholar

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G. Loewenstein and D. Prelec, Anomalies in intertemporal choice: Evidence and an interpretation, The Quarterly Journal of Economics, 107 (1992), 573-597.  doi: 10.2307/2118482.  Google Scholar

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J. MaP. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitly a four step scheme, Probab. Theory Related Fields, 98 (1994), 339-359.  doi: 10.1007/BF01192258.  Google Scholar

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[32]

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[33]

E. S. Phelps and R. A. Pollak, On second-best national saving and game-equilibrium growth, Studies in Macroeconomic Theory, (1980), 201–215. Google Scholar

[34]

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

[35]

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

[36]

Z. Sun and X. Guo, Equilibrium for a time-inconsistent stochastic linear–quadratic control system with jumps and its application to the mean-variance problem, J. Optim. Theory Appl., 181 (2019), 383-410.  doi: 10.1007/s10957-018-01471-x.  Google Scholar

[37]

T. Wang, Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I, Math. Control Relat. Fields, 9 (2019), 385-409.  doi: 10.3934/mcrf.2019018.  Google Scholar

[38]

T. Wang, On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems, ESAIM Control Optim. Calc. Var., 26 (2020), 34pp. doi: 10.1051/cocv/2019057.  Google Scholar

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T. Wang, Equilibrium controls in time inconsistent stochastic linear quadratic problems, Appl. Math. Optim., 81 (2020), 591-619.  doi: 10.1007/s00245-018-9513-x.  Google Scholar

[40]

T. WangZ. Jin and J. Wei, Mean-variance portfolio selection under a non-Markovian regime-switching model: Time-consistent Solutions, SIAM J. Control Optim., 57 (2019), 3249-3271.  doi: 10.1137/18M1186423.  Google Scholar

[41]

H. Wang and Z. Wu, Time-inconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation, Math. Control Relat. Fields, 5 (2015), 651-678.  doi: 10.3934/mcrf.2015.5.651.  Google Scholar

[42]

H. Wang and J. Yong, Time–inconsistent stochastic optimal control problems and backward stochastic Volterra integral equations, ESAIM Control Optim. Calc. Var., 27 (2021), 40pp. doi: 10.1051/cocv/2021027.  Google Scholar

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J. Wei, Time-inconsistent optimal control problems with regime-switching, Math. Control Relat. Fields, 7 (2017), 585-622.  doi: 10.3934/mcrf.2017022.  Google Scholar

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

[45]

W. Yan and J. Yong, Time-inconsistent optimal control problems and related issues, Modeling, Stochastic Control, Optimization, and Applications, 164 (2019), 533-569.   Google Scholar

[46]

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

[47]

J. Yong, Deterministic time-inconsistent optimal control problems-An essentially cooperative approach, Acta Math. Appl. Sin. Engl. Ser., 28 (2012), 1-30.  doi: 10.1007/s10255-012-0120-3.  Google Scholar

[48]

J. Yong, Time-inconsistent optimal control problems and the equilibrium HJB equation, Math. Control Relat. Fields, 2 (2012), 271-329.  doi: 10.3934/mcrf.2012.2.271.  Google Scholar

[49]

J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations–time-consistent solutions, Trans. Amer. Math. Soc., 369 (2017), 5467-5523.  doi: 10.1090/tran/6502.  Google Scholar

[50]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Applications of Mathematics (New York), 43. Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[51]

J. ZhangP. ChenZ. Jin and S. Li, Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model, J. Ind. Manag. Optim., 17 (2021), 765-777.  doi: 10.3934/jimo.2019133.  Google Scholar

[52]

X. ZhangZ. Sun and J. Xiong, A general stochastic maximum principle for a Markov regime switching jump-diffusion model of mean-field type, SIAM J. Control Optim., 56 (2018), 2563-2592.  doi: 10.1137/17M112395X.  Google Scholar

[53]

X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Appl. Math. Optim., 42 (2000), 19-33.  doi: 10.1007/s002450010003.  Google Scholar

show all references

References:
[1]

I. Alia, A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion, Math. Control Relat. Fields, 9 (2019), 541-570.  doi: 10.3934/mcrf.2019025.  Google Scholar

[2]

I. Alia, Time-inconsistent stochastic optimal control problems: A backward stochastic partial differential equations approach, Math. Control Relat. Fields, 10 (2020), 785-826.  doi: 10.3934/mcrf.2020020.  Google Scholar

[3]

I. AliaF. Chighoub and A. Sohail, A characterization of equilibrium strategies in continuous-time mean-variance problems for insurers, Insurance Math. Econom., 68 (2016), 212-223.  doi: 10.1016/j.insmatheco.2016.03.009.  Google Scholar

[4]

T. Björk and A. Murgoci, A general theory of Markovian time-inconsistent stochastic control problems, \emphSSRN, (2010), 55pp, Available from: https://ssrn.com/abstract=1694759. Google Scholar

[5]

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

[6]

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

[7]

S. Christensen and K. Lindensjö, On time-inconsistent stopping problems and mixed strategy stopping times, Stochastic Process. Appl., 130 (2020), 2886-2917.  doi: 10.1016/j.spa.2019.08.010.  Google Scholar

[8]

S. Christensen and K. Lindensjö, Time-inconsistent stopping, myopic adjustment & equilibrium stability: With a mean–variance application, Banach Center Publications, 122 (2020), 53-76.  doi: 10.4064/bc122-4.  Google Scholar

[9]

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

[10]

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

[11]

F. Dou and Q. Lü, Time-inconsistent linear quadratic optimal control problems for stochastic evolution equations, SIAM J. Control Optim., 58 (2020), 485-509.  doi: 10.1137/19M1250339.  Google Scholar

[12]

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

[13]

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

[14]

R. Elie, Contr\^{o}le stochastique et méthodes numériques en finance mathématique, Ph.D. Thesis, University Paris-Dauphine, 2006. Available from: https://pastel.archives-ouvertes.fr/tel-00122883/file/thesis.pdf. Google Scholar

[15]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Inc., Englewood Cliffs, N. J, 1964.  Google Scholar

[16]

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

[17]

Y. Hamaguchi, Small-time solvability of a flow of forward-backward stochastic differential equations, Appl. Math. Optim., 84 (2021), 567-588.  doi: 10.1007/s00245-020-09654-7.  Google Scholar

[18]

Y. Hamaguchi, Time-inconsistent consumption-investment problems in incomplete markets under general discount functions, SIAM J. Control Optim., 59 (2021), 2121-2146.  doi: 10.1137/19M1303782.  Google Scholar

[19]

Y. Hamaguchi, Extended backward stochastic Volterra integral equations and their applications to time-inconsistent stochastic recursive control problems, Math. Control Relat. Fields, 11 (2021), 433-478.  doi: 10.3934/mcrf.2020043.  Google Scholar

[20]

X. He and Z. Jiang, On the equilibrium strategies for time–inconsistent problems in continuous time, preprint. doi: 10.2139/ssrn.3308274.  Google Scholar

[21]

C. Hernández and D. Possamaï, Me, myself and I: A general theory of non–Markovian time–inconsistent stochastic control for sophisticated agents, preprint, arXiv: 2002.12572v2. Google Scholar

[22]

Y. HuH. Jin and X. Y. Zhou, Time-inconsistent stochastic linear quadratic control, SIAM J. Control Optim., 50 (2012), 1548-1572.  doi: 10.1137/110853960.  Google Scholar

[23]

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

[24]

Y. Hu, J. Huang and X. Li, Equilibrium for time-inconsistent stochastic linear–quadratic control under constraint, preprint, arXiv: 1703.09415v1. doi: 10.1137/15M1019040.  Google Scholar

[25]

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

[26]

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

[27]

K. Lindensjö, A regular equilibrium solves the extended HJB system, Oper. Res. Lett., 47 (2019), 427-432.  doi: 10.1016/j.orl.2019.07.011.  Google Scholar

[28]

G. Loewenstein and D. Prelec, Anomalies in intertemporal choice: Evidence and an interpretation, The Quarterly Journal of Economics, 107 (1992), 573-597.  doi: 10.2307/2118482.  Google Scholar

[29]

J. MaP. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitly a four step scheme, Probab. Theory Related Fields, 98 (1994), 339-359.  doi: 10.1007/BF01192258.  Google Scholar

[30]

J. MaJ. Yong and Y. Zhao, Four step scheme for general Markovian forward-backward SDES, J. Syst. Sci. Complex., 23 (2010), 546-571.  doi: 10.1007/s11424-010-0145-8.  Google Scholar

[31]

H. Mei and J. Yong, Equilibrium strategies for time-inconsistent stochastic switching systems, ESAIM Control Optim. Calc. Var., 25 (2019), 60pp. doi: 10.1051/cocv/2018051.  Google Scholar

[32]

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

[33]

E. S. Phelps and R. A. Pollak, On second-best national saving and game-equilibrium growth, Studies in Macroeconomic Theory, (1980), 201–215. Google Scholar

[34]

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

[35]

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

[36]

Z. Sun and X. Guo, Equilibrium for a time-inconsistent stochastic linear–quadratic control system with jumps and its application to the mean-variance problem, J. Optim. Theory Appl., 181 (2019), 383-410.  doi: 10.1007/s10957-018-01471-x.  Google Scholar

[37]

T. Wang, Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I, Math. Control Relat. Fields, 9 (2019), 385-409.  doi: 10.3934/mcrf.2019018.  Google Scholar

[38]

T. Wang, On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems, ESAIM Control Optim. Calc. Var., 26 (2020), 34pp. doi: 10.1051/cocv/2019057.  Google Scholar

[39]

T. Wang, Equilibrium controls in time inconsistent stochastic linear quadratic problems, Appl. Math. Optim., 81 (2020), 591-619.  doi: 10.1007/s00245-018-9513-x.  Google Scholar

[40]

T. WangZ. Jin and J. Wei, Mean-variance portfolio selection under a non-Markovian regime-switching model: Time-consistent Solutions, SIAM J. Control Optim., 57 (2019), 3249-3271.  doi: 10.1137/18M1186423.  Google Scholar

[41]

H. Wang and Z. Wu, Time-inconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation, Math. Control Relat. Fields, 5 (2015), 651-678.  doi: 10.3934/mcrf.2015.5.651.  Google Scholar

[42]

H. Wang and J. Yong, Time–inconsistent stochastic optimal control problems and backward stochastic Volterra integral equations, ESAIM Control Optim. Calc. Var., 27 (2021), 40pp. doi: 10.1051/cocv/2021027.  Google Scholar

[43]

J. Wei, Time-inconsistent optimal control problems with regime-switching, Math. Control Relat. Fields, 7 (2017), 585-622.  doi: 10.3934/mcrf.2017022.  Google Scholar

[44]

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

[45]

W. Yan and J. Yong, Time-inconsistent optimal control problems and related issues, Modeling, Stochastic Control, Optimization, and Applications, 164 (2019), 533-569.   Google Scholar

[46]

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

[47]

J. Yong, Deterministic time-inconsistent optimal control problems-An essentially cooperative approach, Acta Math. Appl. Sin. Engl. Ser., 28 (2012), 1-30.  doi: 10.1007/s10255-012-0120-3.  Google Scholar

[48]

J. Yong, Time-inconsistent optimal control problems and the equilibrium HJB equation, Math. Control Relat. Fields, 2 (2012), 271-329.  doi: 10.3934/mcrf.2012.2.271.  Google Scholar

[49]

J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations–time-consistent solutions, Trans. Amer. Math. Soc., 369 (2017), 5467-5523.  doi: 10.1090/tran/6502.  Google Scholar

[50]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Applications of Mathematics (New York), 43. Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[51]

J. ZhangP. ChenZ. Jin and S. Li, Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model, J. Ind. Manag. Optim., 17 (2021), 765-777.  doi: 10.3934/jimo.2019133.  Google Scholar

[52]

X. ZhangZ. Sun and J. Xiong, A general stochastic maximum principle for a Markov regime switching jump-diffusion model of mean-field type, SIAM J. Control Optim., 56 (2018), 2563-2592.  doi: 10.1137/17M112395X.  Google Scholar

[53]

X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Appl. Math. Optim., 42 (2000), 19-33.  doi: 10.1007/s002450010003.  Google Scholar

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