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Non-exponential discounting portfolio management with habit formation
Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach
Department of Mathematics, University of Bordj Bou Arreridj, 34000 Algeria |
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.
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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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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