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December  2020, 10(4): 785-826. doi: 10.3934/mcrf.2020020

Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach

Ishak Alia

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

Received  June 2019 Revised  December 2019 Published  December 2020 Early access  March 2020

In this paper, we investigate a class of time-inconsistent stochastic control problems for stochastic differential equations with deterministic coefficients. We study these problems within the game theoretic framework, and look for open-loop Nash equilibrium controls. Under suitable conditions, we derive a verification theorem for equilibrium controls via a flow of forward-backward stochastic partial differential equations. To illustrate our results, we discuss a mean-variance problem with a state-dependent trade-off between the mean and the variance.

Keywords: Time-inconsistency, stochastic control, equilibrium control, backward stochastic partial differential equations, mean-variance problem.
Mathematics Subject Classification: Primary: 91B51, 93E20, 60H30, 93E99, 60H10; Secondary: 93E25.
Citation: Ishak Alia. Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach. Mathematical Control & Related Fields, 2020, 10 (4) : 785-826. doi: 10.3934/mcrf.2020020
References:
[1]

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

[2]

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

[3]

D. Andersson and B. Djehiche, A maximum principle for SDEs of mean-field type, Applied Mathematics And Optimization, 63 (2011), 341-356.  doi: 10.1007/s00245-010-9123-8.  Google Scholar

[4]

R. Buckdahn, B. Djehiche and J. Li, A general stochastic maximum principle for SDEs of mean-field type, Applied Mathematics And Optimization, 64 (2011), 197-216.  doi: 10.1007/s00245-011-9136-y.  Google Scholar

[5]

R. Buckdahn, J. Li and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Processes and their Applications, 119 (2009), 3133-3154.  doi: 10.1016/j.spa.2009.05.002.  Google Scholar

[6]

R. Buckdahn and J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part I, Stochastic Processes and their Applications, 93 (2001), 181-204.  doi: 10.1016/S0304-4149(00)00093-4.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

L. Delong, Time-inconsistent stochastic optimal control problems in insurance and finance, Collegium of Economic Analysis Annals, 51 (2018), 229-254.   Google Scholar

[13]

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

[14]

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

[15]

K. Du and Q. Zhang, Semi-linear degenerate backward stochastic partial differential equations and associated forward-backward stochastic differential equations, Stochastic Processes and their Applications, 123 (2013), 1616-1637.  doi: 10.1016/j.spa.2013.01.005.  Google Scholar

[16]

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

[17]

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

[18]

Y. Hamaguchi, Small-time solvability of a flow of forward-backward stochastic differential equations, Applied Mathematics and Optimization (2020), arXiv: 1902.11178v1. Google Scholar

[19]

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

[20]

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

[21]

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

[22]

Y. Hu, J. Ma and J. Yong, On semi-linear degenerate backward stochastic partial differential equations, Probability Theory and Related Fields, 123 (2002), 381-411.  doi: 10.1007/s004400100193.  Google Scholar

[23]

H. Jin and X. Y. Zhou, Behavioral portfolio selection in continuous time, Mathematical Finance, 18 (2008), 385-426.  doi: 10.1111/j.1467-9965.2008.00339.x.  Google Scholar

[24]

C. Karnam, J. Ma and J. Zhang, Dynamic approaches for some time inconsistent problems, Annals of Applied Probability, 27 (2017), 3435-3477.  doi: 10.1214/17-AAP1284.  Google Scholar

[25]

H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, 24. Cambridge University Press, 1990.  Google Scholar

[26]

H. Kunita, Stochastic Flows and Jump-Diffusions, Volume 92 of Probability Theory and Stochastic Modelling. Springer, Singapore, 2019. doi: 10.1007/978-981-13-3801-4.  Google Scholar

[27]

H. Kunita, Some extensions of Itô's formula, Séminaire de Probabilités XV 1979/80, 118–141, Lecture Notes in Math., 850, Springer, Berlin, 1981.  Google Scholar

[28]

D. Li and W. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Mathematical Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100.  Google Scholar

[29]

J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications, Stochastic Processes and their Applications, 70 (1997), 59-84.  doi: 10.1016/S0304-4149(97)00057-4.  Google Scholar

[30]

J. Ma and J. Yong, On linear degenerate backward stochastic partial differential equations, Probability Theory and Related Fields, 113 (1999), 135-170.  doi: 10.1007/s004400050205.  Google Scholar

[31]

J. Ma, H. Yin and J. F. Zhang, On non-Markovian forward-backward SDEs and backward stochastic PDEs, Stochastic Processes and their Applications, 122 (2012), 3980-4004.  doi: 10.1016/j.spa.2012.08.002.  Google Scholar

[32]

H. Mei and J. Yong, Equilibrium strategies for time-inconsistent stochastic switching systems, ESAIM: Control, Optimisation and Calculus of Variations, 25 (2019), Art. 64, 60 pp. doi: 10.1051/cocv/2018051.  Google Scholar

[33]

D. Ocone and E. Pardoux, A generalized Itô–Ventzell formula. Application to a class of anticipating stochastic differential equations, Annales de l'I. H. P., Section B, 25 (1989), 39-71.   Google Scholar

[34]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM Journal on Control and Optimization, 28 (1990), 966-979.  doi: 10.1137/0328054.  Google Scholar

[35]

S. Peng, Maximum principle for stochastic optimal control with non convex control domain, Lecture Notes in Control & Information Sciences, 114 (1990), 724-732.  doi: 10.1007/BFb0120094.  Google Scholar

[36]

H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, , Volume 61 of Stochastic Modelling and Applied Probability. Springer, Berlin Heidelberg, 2009. doi: 10.1007/978-3-540-89500-8.  Google Scholar

[37]

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

[38]

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

[39]

T. Wang, Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I, Math. Control Relat. Fields, 9 (2019), 385–409, arXiv: 1802.01080v1. doi: 10.3934/mcrf.2019018.  Google Scholar

[40]

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

[41]

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

[42]

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

[43]

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

[44]

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

[45]

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

[46]

Y. Zeng, Z. F. Li and Y. Z. Lai, Time-consistent investment and reinsurance strategies for mean–variance insurers with jumps, Insurance: Mathematics and Economics, 52 (2013), 498-507.  doi: 10.1016/j.insmatheco.2013.02.007.  Google Scholar

[47]

X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics And Optimization, 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, Mathematical Control & Related Fields, 9 (2019), 541-570.  doi: 10.3934/mcrf.2019025.  Google Scholar

[2]

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

[3]

D. Andersson and B. Djehiche, A maximum principle for SDEs of mean-field type, Applied Mathematics And Optimization, 63 (2011), 341-356.  doi: 10.1007/s00245-010-9123-8.  Google Scholar

[4]

R. Buckdahn, B. Djehiche and J. Li, A general stochastic maximum principle for SDEs of mean-field type, Applied Mathematics And Optimization, 64 (2011), 197-216.  doi: 10.1007/s00245-011-9136-y.  Google Scholar

[5]

R. Buckdahn, J. Li and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Processes and their Applications, 119 (2009), 3133-3154.  doi: 10.1016/j.spa.2009.05.002.  Google Scholar

[6]

R. Buckdahn and J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part I, Stochastic Processes and their Applications, 93 (2001), 181-204.  doi: 10.1016/S0304-4149(00)00093-4.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

L. Delong, Time-inconsistent stochastic optimal control problems in insurance and finance, Collegium of Economic Analysis Annals, 51 (2018), 229-254.   Google Scholar

[13]

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

[14]

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

[15]

K. Du and Q. Zhang, Semi-linear degenerate backward stochastic partial differential equations and associated forward-backward stochastic differential equations, Stochastic Processes and their Applications, 123 (2013), 1616-1637.  doi: 10.1016/j.spa.2013.01.005.  Google Scholar

[16]

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

[17]

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

[18]

Y. Hamaguchi, Small-time solvability of a flow of forward-backward stochastic differential equations, Applied Mathematics and Optimization (2020), arXiv: 1902.11178v1. Google Scholar

[19]

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

[20]

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

[21]

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

[22]

Y. Hu, J. Ma and J. Yong, On semi-linear degenerate backward stochastic partial differential equations, Probability Theory and Related Fields, 123 (2002), 381-411.  doi: 10.1007/s004400100193.  Google Scholar

[23]

H. Jin and X. Y. Zhou, Behavioral portfolio selection in continuous time, Mathematical Finance, 18 (2008), 385-426.  doi: 10.1111/j.1467-9965.2008.00339.x.  Google Scholar

[24]

C. Karnam, J. Ma and J. Zhang, Dynamic approaches for some time inconsistent problems, Annals of Applied Probability, 27 (2017), 3435-3477.  doi: 10.1214/17-AAP1284.  Google Scholar

[25]

H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, 24. Cambridge University Press, 1990.  Google Scholar

[26]

H. Kunita, Stochastic Flows and Jump-Diffusions, Volume 92 of Probability Theory and Stochastic Modelling. Springer, Singapore, 2019. doi: 10.1007/978-981-13-3801-4.  Google Scholar

[27]

H. Kunita, Some extensions of Itô's formula, Séminaire de Probabilités XV 1979/80, 118–141, Lecture Notes in Math., 850, Springer, Berlin, 1981.  Google Scholar

[28]

D. Li and W. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Mathematical Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100.  Google Scholar

[29]

J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications, Stochastic Processes and their Applications, 70 (1997), 59-84.  doi: 10.1016/S0304-4149(97)00057-4.  Google Scholar

[30]

J. Ma and J. Yong, On linear degenerate backward stochastic partial differential equations, Probability Theory and Related Fields, 113 (1999), 135-170.  doi: 10.1007/s004400050205.  Google Scholar

[31]

J. Ma, H. Yin and J. F. Zhang, On non-Markovian forward-backward SDEs and backward stochastic PDEs, Stochastic Processes and their Applications, 122 (2012), 3980-4004.  doi: 10.1016/j.spa.2012.08.002.  Google Scholar

[32]

H. Mei and J. Yong, Equilibrium strategies for time-inconsistent stochastic switching systems, ESAIM: Control, Optimisation and Calculus of Variations, 25 (2019), Art. 64, 60 pp. doi: 10.1051/cocv/2018051.  Google Scholar

[33]

D. Ocone and E. Pardoux, A generalized Itô–Ventzell formula. Application to a class of anticipating stochastic differential equations, Annales de l'I. H. P., Section B, 25 (1989), 39-71.   Google Scholar

[34]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM Journal on Control and Optimization, 28 (1990), 966-979.  doi: 10.1137/0328054.  Google Scholar

[35]

S. Peng, Maximum principle for stochastic optimal control with non convex control domain, Lecture Notes in Control & Information Sciences, 114 (1990), 724-732.  doi: 10.1007/BFb0120094.  Google Scholar

[36]

H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, , Volume 61 of Stochastic Modelling and Applied Probability. Springer, Berlin Heidelberg, 2009. doi: 10.1007/978-3-540-89500-8.  Google Scholar

[37]

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

[38]

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

[39]

T. Wang, Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I, Math. Control Relat. Fields, 9 (2019), 385–409, arXiv: 1802.01080v1. doi: 10.3934/mcrf.2019018.  Google Scholar

[40]

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

[41]

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

[42]

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

[43]

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

[44]

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

[45]

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

[46]

Y. Zeng, Z. F. Li and Y. Z. Lai, Time-consistent investment and reinsurance strategies for mean–variance insurers with jumps, Insurance: Mathematics and Economics, 52 (2013), 498-507.  doi: 10.1016/j.insmatheco.2013.02.007.  Google Scholar

[47]

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

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