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

Stochastic global maximum principle for optimization with recursive utilities

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

Received  September 19, 2016 Revised  January 05, 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.
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]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[2]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[3]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[4]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[5]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192

[6]

Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021009

[7]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008

[8]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088

[9]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827

[10]

Xiaohu Wang, Dingshi Li, Jun Shen. Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2829-2855. doi: 10.3934/dcdsb.2020207

[11]

Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021012

[12]

Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208

[13]

Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021007

[14]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[15]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203

[16]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023

[17]

Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201

[18]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066

[19]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183

[20]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[Back to Top]