• Previous Article
    On optimal $ L^1 $-control in coefficients for quasi-linear Dirichlet boundary value problems with $ BMO $-anisotropic $ p $-Laplacian
  • MCRF Home
  • This Issue
  • Next Article
    Non-exponential discounting portfolio management with habit formation
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. AliaF. 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. BuckdahnB. 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. BuckdahnJ. 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. BjörkA. 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. BjorkM. 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. HuH. 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. HuH. 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. HuJ. 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. KarnamJ. 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. MaH. 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. WeiJ. 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. ZengZ. 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. AliaF. 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. BuckdahnB. 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. BuckdahnJ. 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. BjörkA. 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. BjorkM. 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. HuH. 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. HuH. 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. HuJ. 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. KarnamJ. 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. MaH. 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. WeiJ. 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. ZengZ. 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.

[1]

Zhiping Chen, Jia Liu, Gang Li. Time consistent policy of multi-period mean-variance problem in stochastic markets. Journal of Industrial and Management Optimization, 2016, 12 (1) : 229-249. doi: 10.3934/jimo.2016.12.229

[2]

Lihua Bian, Zhongfei Li, Haixiang Yao. Time-consistent strategy for a multi-period mean-variance asset-liability management problem with stochastic interest rate. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1383-1410. doi: 10.3934/jimo.2020026

[3]

Huai-Nian Zhu, Cheng-Ke Zhang, Zhuo Jin. Continuous-time mean-variance asset-liability management with stochastic interest rates and inflation risks. Journal of Industrial and Management Optimization, 2020, 16 (2) : 813-834. doi: 10.3934/jimo.2018180

[4]

Yingxu Tian, Junyi Guo, Zhongyang Sun. Optimal mean-variance reinsurance in a financial market with stochastic rate of return. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1887-1912. doi: 10.3934/jimo.2020051

[5]

Liyuan Wang, Zhiping Chen, Peng Yang. Robust equilibrium control-measure policy for a DC pension plan with state-dependent risk aversion under mean-variance criterion. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1203-1233. doi: 10.3934/jimo.2020018

[6]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control and Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97

[7]

Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021020

[8]

Yushi Hamaguchi. Extended backward stochastic Volterra integral equations and their applications to time-Inconsistent stochastic recursive control problems. Mathematical Control and Related Fields, 2021, 11 (2) : 433-478. doi: 10.3934/mcrf.2020043

[9]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control and Related Fields, 2021, 11 (4) : 797-828. doi: 10.3934/mcrf.2020047

[10]

Hao Chang, Jiaao Li, Hui Zhao. Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1393-1423. doi: 10.3934/jimo.2021025

[11]

Yinggu Chen, Said HamadÈne, Tingshu Mu. Mean-field doubly reflected backward stochastic differential equations. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022012

[12]

Haiyang Wang, Zhen Wu. Time-inconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation. Mathematical Control and Related Fields, 2015, 5 (3) : 651-678. doi: 10.3934/mcrf.2015.5.651

[13]

Shuang Li, Chuong Luong, Francisca Angkola, Yonghong Wu. Optimal asset portfolio with stochastic volatility under the mean-variance utility with state-dependent risk aversion. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1521-1533. doi: 10.3934/jimo.2016.12.1521

[14]

Haixiang Yao, Zhongfei Li, Yongzeng Lai. Dynamic mean-variance asset allocation with stochastic interest rate and inflation rate. Journal of Industrial and Management Optimization, 2016, 12 (1) : 187-209. doi: 10.3934/jimo.2016.12.187

[15]

Qian Zhao, Yang Shen, Jiaqin Wei. Mean-variance investment and contribution decisions for defined benefit pension plans in a stochastic framework. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1147-1171. doi: 10.3934/jimo.2020015

[16]

Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879

[17]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control and Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613

[18]

Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations and Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035

[19]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control and Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401

[20]

Chuchu Chen, Jialin Hong. Mean-square convergence of numerical approximations for a class of backward stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 2051-2067. doi: 10.3934/dcdsb.2013.18.2051

2020 Impact Factor: 1.284

Metrics

  • PDF downloads (483)
  • HTML views (474)
  • Cited by (0)

Other articles
by authors

[Back to Top]