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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

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) Google Scholar

[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) Google Scholar

[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) Google Scholar

[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) Google Scholar

[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) Google Scholar

[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 Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

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) Google Scholar

[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) Google Scholar

[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) Google Scholar

[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) Google Scholar

[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) Google Scholar

[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 Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

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