American Institute of Mathematical Sciences

Advanced
January  2018, 3: 8 doi: 10.1186/s41546-018-0033-z

Optimal control with delayed information flow of systems driven by G-Brownian motion

Francesca Biagini 1,2,, , Thilo Meyer-Brandis 3, , Bernt Øksendal 2, and  Krzysztof Paczka 2,

1. Department of Mathematics, LMU Munich, Theresienstraße 39, 80333 Munich, Germany;

2. Department of Mathematics, University of Oslo, P. O. Box 1053 Blindern, N-0316 Oslo, Norway;

3. Department of Mathematics, University of Munich, Theresienstraße 39, 80333 Munich, Germany

Received  September 14, 2017 Revised  September 19, 2018 Published  October 2018

In this paper, we study strongly robust optimal control problems under volatility uncertainty. In the G-framework, we adapt the stochastic maximum principle to find necessary and sufficient conditions for the existence of a strongly robust optimal control.
Keywords: G-Brownian motion, optimal control problem, stochastic maximum principle.
Mathematics Subject Classification: 60H99;93E20.
Citation: Francesca Biagini, Thilo Meyer-Brandis, Bernt Øksendal, Krzysztof Paczka. Optimal control with delayed information flow of systems driven by G-Brownian motion. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 8-. doi: 10.1186/s41546-018-0033-z
References:
[1]

Denis, L., Hu, M., Peng, S.:Function spaces and capacity related to a sublinear expectation:application to G-Brownian motion paths. Potential Anal. 34, 139-161 (2011)

[2]

Hu, M., Ji, S.:Stochastic maximum principle for stochastic recursive optimal control problem under volatility ambiguity. SIAM J. Control Optim. 54(2), 918-945 (2016)

[3]

Hu, M., Ji, S., Peng, S.:Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion. Stoch. Process. Appl. 124, 1170-1195 (2014a)

[4]

Hu, M., Ji, S., Yang, S.:A stochastic recursive optimal control problem under the G-expectation framework. Appl. Math. Optim. 70, 253-278 (2014b)

[5]

Hu, M., Ji, S., Peng, S., Song, Y.:Backward stochastic differential equations driven by G-Brownian motion. Stoch. Process. Appl. 124, 759-784 (2014c)

[6]

Matoussi, A., Possamai, D., Zhou, C.:Robust utility maximization in non-dominated models with 2BSDEs. Math. Financ. (2013). https://doi.org/10.1111/mafi.12031

[7]

Peng, S.:G-expectation, G-Brownian motion and related stochastic calculus of Itô type. Stoch. Anal. Appl. 2, 541-567 (2007)

[8]

Peng, S.:Nonlinear expectations and stochastic calculus under uncertainty. Preprint, arXiv1002.4546v1(2010)

[9]

Peng, S., Song, Y., Zhang, J.:A complete representation theorem for G-martingales. Stochastics. 86, 609-631 (2014)

[10]

Soner, M., Touzi, N., Zhang, J.:Martingale representation theorem for the G-expectation. Stoch. Anal. Appl. 121, 265-287 (2011a)

[11]

Soner, M., Touzi, N., Zhang, J.:Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16, 1844-1879 (2011b)

[12]

Soner, M., Touzi, N., Zhang, J.:Wellposedness of second order backward SDEs. Probab. Theory Relat. Fields. 153, 149-190 (2011c)

[13]

Song, Y.:Some properties on G-evaluation and its applications to G-martingale decomposition. Sci. China. 54, 287-300 (2011)

[14]

Song, Y.:Uniqueness of the representation for G-martingales with finite variation. Electron. J. Probab. 17, 1-15 (2012)

show all references

References:
[1]

Denis, L., Hu, M., Peng, S.:Function spaces and capacity related to a sublinear expectation:application to G-Brownian motion paths. Potential Anal. 34, 139-161 (2011)

[2]

Hu, M., Ji, S.:Stochastic maximum principle for stochastic recursive optimal control problem under volatility ambiguity. SIAM J. Control Optim. 54(2), 918-945 (2016)

[3]

Hu, M., Ji, S., Peng, S.:Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion. Stoch. Process. Appl. 124, 1170-1195 (2014a)

[4]

Hu, M., Ji, S., Yang, S.:A stochastic recursive optimal control problem under the G-expectation framework. Appl. Math. Optim. 70, 253-278 (2014b)

[5]

Hu, M., Ji, S., Peng, S., Song, Y.:Backward stochastic differential equations driven by G-Brownian motion. Stoch. Process. Appl. 124, 759-784 (2014c)

[6]

Matoussi, A., Possamai, D., Zhou, C.:Robust utility maximization in non-dominated models with 2BSDEs. Math. Financ. (2013). https://doi.org/10.1111/mafi.12031

[7]

Peng, S.:G-expectation, G-Brownian motion and related stochastic calculus of Itô type. Stoch. Anal. Appl. 2, 541-567 (2007)

[8]

Peng, S.:Nonlinear expectations and stochastic calculus under uncertainty. Preprint, arXiv1002.4546v1(2010)

[9]

Peng, S., Song, Y., Zhang, J.:A complete representation theorem for G-martingales. Stochastics. 86, 609-631 (2014)

[10]

Soner, M., Touzi, N., Zhang, J.:Martingale representation theorem for the G-expectation. Stoch. Anal. Appl. 121, 265-287 (2011a)

[11]

Soner, M., Touzi, N., Zhang, J.:Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16, 1844-1879 (2011b)

[12]

Soner, M., Touzi, N., Zhang, J.:Wellposedness of second order backward SDEs. Probab. Theory Relat. Fields. 153, 149-190 (2011c)

[13]

Song, Y.:Some properties on G-evaluation and its applications to G-martingale decomposition. Sci. China. 54, 287-300 (2011)

[14]

Song, Y.:Uniqueness of the representation for G-martingales with finite variation. Electron. J. Probab. 17, 1-15 (2012)

[1]

Zhengyan Lin, Li-Xin Zhang. Convergence to a self-normalized G-Brownian motion. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 4-. doi: 10.1186/s41546-017-0013-8
[2]

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
[3]

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
[4]

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
[5]

Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281
[6]

Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1197-1212. doi: 10.3934/dcdsb.2014.19.1197
[7]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control and Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018
[8]

Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157
[9]

Yong Ren, Huijin Yang, Wensheng Yin. Weighted exponential stability of stochastic coupled systems on networks with delay driven by $ G $-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3379-3393. doi: 10.3934/dcdsb.2018325
[10]

Dingjun Yao, Rongming Wang, Lin Xu. Optimal asset control of a geometric Brownian motion with the transaction costs and bankruptcy permission. Journal of Industrial and Management Optimization, 2015, 11 (2) : 461-478. doi: 10.3934/jimo.2015.11.461
[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]

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 and Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[13]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control and Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195
[14]

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
[15]

Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199
[16]

Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174
[17]

Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161
[18]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations and Control Theory, 2022, 11 (2) : 347-371. doi: 10.3934/eect.2020110
[19]

Miaomiao Chen, Rong Yuan. Maximum principle for the optimal harvesting problem of a size-stage-structured population model. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021245
[20]

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

 Impact Factor: 

Tools

Metrics

  • PDF downloads (28)
  • HTML views (283)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy