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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

 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.

Citation: Ishak Alia. Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach. Mathematical Control and 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. [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. [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. [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. [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. [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. [7] S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Review of Financial Studies, 23 (2010), 2970-3016. [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. [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. [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. [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. [12] L. Delong, Time-inconsistent stochastic optimal control problems in insurance and finance, Collegium of Economic Analysis Annals, 51 (2018), 229-254. [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. [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. [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. [16] I. Ekeland and A. Lazrak, Equilibrium policies when preferences are time-inconsistent, preprint, arXiv: 0808.3790v1. [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. [18] Y. Hamaguchi, Small-time solvability of a flow of forward-backward stochastic differential equations, Applied Mathematics and Optimization (2020), arXiv: 1902.11178v1. [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. [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. [21] Y. Hu, J. Huang and X. Li, Equilibrium for time-inconsistent stochastic linear–quadratic control under constraint, preprint, arXiv: 1703.09415v1. [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. [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. [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. [25] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, 24. Cambridge University Press, 1990. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [40] J. Wei, Time-inconsistent optimal control problems with regime-switching, Mathematical Control & Related Fields, 7 (2017), 585-622.  doi: 10.3934/mcrf.2017022. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [7] S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Review of Financial Studies, 23 (2010), 2970-3016. [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. [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. [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. [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. [12] L. Delong, Time-inconsistent stochastic optimal control problems in insurance and finance, Collegium of Economic Analysis Annals, 51 (2018), 229-254. [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. [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. [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. [16] I. Ekeland and A. Lazrak, Equilibrium policies when preferences are time-inconsistent, preprint, arXiv: 0808.3790v1. [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. [18] Y. Hamaguchi, Small-time solvability of a flow of forward-backward stochastic differential equations, Applied Mathematics and Optimization (2020), arXiv: 1902.11178v1. [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. [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. [21] Y. Hu, J. Huang and X. Li, Equilibrium for time-inconsistent stochastic linear–quadratic control under constraint, preprint, arXiv: 1703.09415v1. [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. [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. [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. [25] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, 24. Cambridge University Press, 1990. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [40] J. Wei, Time-inconsistent optimal control problems with regime-switching, Mathematical Control & Related Fields, 7 (2017), 585-622.  doi: 10.3934/mcrf.2017022. [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. [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. [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. [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. [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. [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. [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.
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