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

Ishak Alia

doi:10.3934/mcrf.2020020

Article Contents

2020, Volume 10, Issue 4: 785-826. Doi: 10.3934/mcrf.2020020

This issue Previous Article Non-exponential discounting portfolio management with habit formation Next Article On optimal

$ L^1 $

-control in coefficients for quasi-linear Dirichlet boundary value problems with

$ BMO $

-anisotropic

$ p $

-Laplacian

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

Early access: March 2020

Published: October 2020

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

Full Text(HTML)

Related Papers

Cited by

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.