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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 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)

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