American Institute of Mathematical Sciences

Advanced
January  2017, 2: 1 doi: 10.1186/s41546-017-0014-7

Stochastic global maximum principle for optimization with recursive utilities

Mingshang Hu ,

Zhongtai Institute of Finance, Shandong University, Jinan, Shandong 250100, People's Republic of China

Received  September 19, 2016 Revised  January 05, 2017 Published  March 2017

Fund Project: supported by NSF (No. 11671231, 11201262 and 10921101), Shandong Province (No.BS2013SF020 and ZR2014AP005), Young Scholars Program of Shandong University and the 111 Project (No. B12023).

In this paper, we study the recursive stochastic optimal control problems. The control domain does not need to be convex, and the generator of the backward stochastic differential equation can contain z. We obtain the variational equations for backward stochastic differential equations, and then obtain the maximum principle which solves completely Peng's open problem.
Keywords: Backward stochastic differential equations, Recursive stochastic optimal control, Maximum principle, Variational equation.
Mathematics Subject Classification: 93E20;60H10;49K45.
Citation: Mingshang Hu. Stochastic global maximum principle for optimization with recursive utilities. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 1-. doi: 10.1186/s41546-017-0014-7
References:
[1]

Briand, PH, Delyon, B, Hu, Y, Pardoux, E, Stoica, L:Lp solutions of backward stochastic differential equations. Stochastic Process. Appl. 108, 109-129 (2003)

[2]

Chen, Z, Epstein, L:Ambiguity, risk, and asset returns in continuous time. Econometrica 70, 1403-1443(2002)

[3]

Dokuchaev, M, Zhou, XY:Stochastic controls with terminal contingent conditions. J. Math. Anal. Appl. 238, 143-165 (1999)

[4]

Duffie, D, Epstein, L:Stochastic differential utility. Econometrica 60, 353-394 (1992)

[5]

El Karoui, N, Peng, S, Quenez, MC:Backward stochastic differential equations in finance. Math. Finance 7, 1-71 (1997)

[6]

El Karoui, N, Peng, S, Quenez, MC:A dynamic maximum priciple for the optimization of recursive utilities under constraints. Ann. Appl. Probab. 11, 664-693 (2001)

[7]

Ji, S, Zhou, XY:A maximum principle for stochastic optimal control with terminal state constrains, and its applications. Comm. Inf. Syst. 6, 321-337 (2006)

[8]

Kohlmann, M, Zhou, XY:Relationship between backward stochastic differential equations and stochastic controls:a linear-quadratic approach. SIAM J. Control Optim. 38, 1392-1407 (2000)

[9]

Lim, A, Zhou, XY:Linear-quadratic control of backward stochastic differential equations. SIAM J.Control Optim. 40, 450-474 (2001)

[10]

Pardoux, E, Peng, S:Adapted Solutions of Backward Stochastic Equations. Systerm Control Lett. 14, 55-61 (1990)

[11]

Peng, S:A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28, 966-979 (1990)

[12]

Peng, S:Backward stochastic differential equations and applications to optimal control. Appl. Math.Optim. 27, 125-144 (1993)

[13]

Peng, S:Open problems on backward stochastic differential equations. In:Chen, S, Li, X, Yong, J, Zhou, XY (eds.) Control of distributed parameter and stocastic systems, pp. 265-273, Boston:Kluwer Acad.Pub. (1998)

[14]

Shi, J, Wu, Z:The maximum principle for fully coupled forward-backward stochastic control system. Acta Automat. Sinica 32, 161-169 (2006)

[15]

Tang, S, Li, X:Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32, 1447-1475 (1994)

[16]

Wu, Z:Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems. Syst. Sci. Math. Sci. 11, 249-259 (1998)

[17]

Wu, Z:A general maximum principle for optimal control of forward-backward stochastic systems.Automatica 49, 1473-1480 (2013)

[18]

Xu, W:Stochastic maximum principle for optimal control problem of forward and backward system. J.Austral. Math. Soc. Ser. B 37, 172-185 (1995)

[19]

Yong, J:Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions. SIAM J. Control Optim. 48, 4119-4156 (2010)

[20]

Yong, J, Zhou, XY:Stochastic controls:Hamiltonian systems and HJB equations. Springer-Verlag, New York (1999)

show all references

References:
[1]

Briand, PH, Delyon, B, Hu, Y, Pardoux, E, Stoica, L:Lp solutions of backward stochastic differential equations. Stochastic Process. Appl. 108, 109-129 (2003)

[2]

Chen, Z, Epstein, L:Ambiguity, risk, and asset returns in continuous time. Econometrica 70, 1403-1443(2002)

[3]

Dokuchaev, M, Zhou, XY:Stochastic controls with terminal contingent conditions. J. Math. Anal. Appl. 238, 143-165 (1999)

[4]

Duffie, D, Epstein, L:Stochastic differential utility. Econometrica 60, 353-394 (1992)

[5]

El Karoui, N, Peng, S, Quenez, MC:Backward stochastic differential equations in finance. Math. Finance 7, 1-71 (1997)

[6]

El Karoui, N, Peng, S, Quenez, MC:A dynamic maximum priciple for the optimization of recursive utilities under constraints. Ann. Appl. Probab. 11, 664-693 (2001)

[7]

Ji, S, Zhou, XY:A maximum principle for stochastic optimal control with terminal state constrains, and its applications. Comm. Inf. Syst. 6, 321-337 (2006)

[8]

Kohlmann, M, Zhou, XY:Relationship between backward stochastic differential equations and stochastic controls:a linear-quadratic approach. SIAM J. Control Optim. 38, 1392-1407 (2000)

[9]

Lim, A, Zhou, XY:Linear-quadratic control of backward stochastic differential equations. SIAM J.Control Optim. 40, 450-474 (2001)

[10]

Pardoux, E, Peng, S:Adapted Solutions of Backward Stochastic Equations. Systerm Control Lett. 14, 55-61 (1990)

[11]

Peng, S:A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28, 966-979 (1990)

[12]

Peng, S:Backward stochastic differential equations and applications to optimal control. Appl. Math.Optim. 27, 125-144 (1993)

[13]

Peng, S:Open problems on backward stochastic differential equations. In:Chen, S, Li, X, Yong, J, Zhou, XY (eds.) Control of distributed parameter and stocastic systems, pp. 265-273, Boston:Kluwer Acad.Pub. (1998)

[14]

Shi, J, Wu, Z:The maximum principle for fully coupled forward-backward stochastic control system. Acta Automat. Sinica 32, 161-169 (2006)

[15]

Tang, S, Li, X:Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32, 1447-1475 (1994)

[16]

Wu, Z:Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems. Syst. Sci. Math. Sci. 11, 249-259 (1998)

[17]

Wu, Z:A general maximum principle for optimal control of forward-backward stochastic systems.Automatica 49, 1473-1480 (2013)

[18]

Xu, W:Stochastic maximum principle for optimal control problem of forward and backward system. J.Austral. Math. Soc. Ser. B 37, 172-185 (1995)

[19]

Yong, J:Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions. SIAM J. Control Optim. 48, 4119-4156 (2010)

[20]

Yong, J, Zhou, XY:Stochastic controls:Hamiltonian systems and HJB equations. Springer-Verlag, New York (1999)

[1]

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
[2]

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
[3]

Zhen Wu, Feng Zhang. Maximum principle for discrete-time stochastic optimal control problem and stochastic game. Mathematical Control and Related Fields, 2022, 12 (2) : 475-493. doi: 10.3934/mcrf.2021031
[4]

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
[5]

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
[6]

Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2289-2331. doi: 10.3934/cpaa.2020100
[7]

Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control and Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022
[8]

Yuanzhuo Song, Zhen Wu. The general maximum principle for stochastic control problems with singular controls. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022106
[9]

Jingtao Shi, Juanjuan Xu, Huanshui Zhang. Stochastic recursive optimal control problem with time delay and applications. Mathematical Control and Related Fields, 2015, 5 (4) : 859-888. doi: 10.3934/mcrf.2015.5.859
[10]

Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial and Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067
[11]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581
[12]

Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499
[13]

Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075
[14]

Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803
[15]

Joscha Diehl, Jianfeng Zhang. Backward stochastic differential equations with Young drift. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 5-. doi: 10.1186/s41546-017-0016-5
[16]

Ștefana-Lucia Aniţa. Optimal control for stochastic differential equations and related Kolmogorov equations. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022023
[17]

Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems and Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002
[18]

Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115
[19]

Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447
[20]

Dejian Chang, Zhen Wu. Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance. Journal of Industrial and Management Optimization, 2015, 11 (1) : 27-40. doi: 10.3934/jimo.2015.11.27

 Impact Factor: 

Tools

Metrics

  • PDF downloads (49)
  • HTML views (651)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy