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Reflected solutions of backward stochastic differential equations driven by G-Brownian motion

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Abstract

In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by G-Brownian motion. The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected G-BSDEs, we apply a “martingale condition” instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization. We then give some applications including a generalized Feynman-Kac formula of an obstacle problem for fully nonlinear partial differential equation and option pricing of American types under volatility uncertainty.

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References

  1. Crandall M, Ishii H, Lions P L. User’s guide to the viscosity solutions of second order partial differential equations. Bull Amer Math Soc (NS), 1992, 27: 1–67

    Article  MathSciNet  MATH  Google Scholar 

  2. Cvitanic J, Karatzas I. Backward stochastic differential equations with reflection and Dynkin games. Ann Probab, 1996, 24: 2024–2056

    Article  MathSciNet  MATH  Google Scholar 

  3. Denis L, Hu M, Peng S. Function spaces and capacity related to a sublinear expectation: Application to G-Brownian motion pathes. Potential Anal, 2011, 34: 139–161

    Article  MathSciNet  MATH  Google Scholar 

  4. El Karoui N, Kapoudjian C, Pardoux E, et al. Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. Ann Probab, 1997, 23: 702–737

    MathSciNet  MATH  Google Scholar 

  5. El Karoui N, Pardoux E, Quenez M C. Reflected backward SDE’s and American options. In: Numerical Methods in Finance. Cambridge: Cambridge University Press, 1997: 215–231

    Chapter  Google Scholar 

  6. Hamadene S. Reflected BSDE’s with discontinuous barrier and application. Stochastics Stochastics Rep, 2002, 74: 571–596

    Article  MathSciNet  MATH  Google Scholar 

  7. Hamadene S, Lepeltier J-P. Reflected BSDE’s and mixed game problem. Stochastic Process Appl, 2000, 85: 177–188

    Article  MathSciNet  MATH  Google Scholar 

  8. Hu M, Ji S, Peng S, et al. Backward stochastic differential equations driven by G-Brownian motion. Stochastic Process Appl, 2014, 124: 759–784

    Article  MathSciNet  MATH  Google Scholar 

  9. Hu M, Ji S, Peng S, et al. Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion. Stochastic Process Appl, 2014, 124: 1170–1195

    Article  MathSciNet  MATH  Google Scholar 

  10. Hu M, Peng S. On representation theorem of G-expectations and paths of G-Brownian motion. Acta Math Appl Sin Engl Ser, 2009, 25: 539–546

    Article  MathSciNet  MATH  Google Scholar 

  11. Hu M, Peng S. Extended conditional G-expectation and related stopping times. ArXiv:1309.3829v1, 2013

    Google Scholar 

  12. Karatzas I. On the pricing of American options. Appl Math Optim, 1988, 17: 37–60

    Article  MathSciNet  MATH  Google Scholar 

  13. Kobylanski M, Lepeltier J-P, Quenez M C, et al. Reflected BSDE with superlinear quadratic coefficient. Probab Math Statist, 2002, 22: 51–83

    MathSciNet  MATH  Google Scholar 

  14. Lepeltier J-P, Xu M. Penalization method for reflected backward stochastic differential equations with one r.c.l.l. barrier. Statist Probab Lett, 2005, 75: 58–66

    Article  MathSciNet  MATH  Google Scholar 

  15. Li X, Peng S. Stopping times and related Itô’s calculus with G-Brownian motion. Stochastic Process Appl, 2011, 121: 1492–1508

    Article  MathSciNet  MATH  Google Scholar 

  16. Li H, Peng S, Song Y. Supermartingale decomposition theorem under G-expectation. ArXiv:1703.02730, 2017

    Google Scholar 

  17. Matoussi A. Reflected solutions of backward stochasitc differential equations with continous coefficient. Statist Probab Lett, 1997, 34: 347–354

    Article  MathSciNet  MATH  Google Scholar 

  18. Peng S. Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type. Probab Theory Related Fields, 1999, 113: 473–499

    Article  MathSciNet  MATH  Google Scholar 

  19. Peng S. Filtration consistent nonlinear expectations and evaluations of contigent claims. Acta Math Appl Sin, 2004, 20: 1–24

    Article  Google Scholar 

  20. Peng S. Nonlinear expecations and nonlinear Markov chains. Chin Ann Math Ser B, 2005, 26: 159–184

    Article  Google Scholar 

  21. Peng S. G-expectation, G-Brownian motion and related stochastic calculus of Itô type. In: Stochastic Analysis and Applications. Abel Symposia, vol. 2. Berlin: Springer, 2007: 541–567

    Article  MATH  Google Scholar 

  22. Peng S. Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stochastic Process Appl, 2008, 118: 2223–2253

    Article  MathSciNet  MATH  Google Scholar 

  23. Peng S. Nonlinear expectations and stochastic calculus under uncertainty. ArXiv:1002.4546v1, 2010

    Google Scholar 

  24. Rodrigues J F. Obstacle Problems in Mathematical Physics. North-Holland Mathematics Studies, vol. 134. Notas de Matematica, vol. 114. Amsterdam: North-Holland, 1987

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Vorbrink J. Financial markets with volatility uncertainty. J Math Econom, 2014, 53: 64–78

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the Tian Yuan Projection of the National Natural Science Foundation of China (Grant Nos. 11526205 and 11626247), the German Research Foundation (DFG) via CRC 1283 and the Lebesgue Center of Mathematics (“Investissements d’aveni” Program) (Grant No. ANR-11-LABX-0020-01). The authors thank Yongsheng Song and Falei Wang for their help and many useful discussions. The authors are grateful to the anonymous referees for their careful reading of the manuscript and many helpful suggestions.

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Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    Hanwu Li & Shige Peng

  2. Zhongtai Institute of Finance, Shandong University, Jinan, 250100, China

    Shige Peng

  3. Institut de Recherche Mathématiques de Rennes, Université de Rennes 1, Rennes Cedex, 35042, France

    Abdoulaye Soumana Hima

  4. Département de Mathématiques, Université de Maradi, Maradi, BP 465, Niger

    Abdoulaye Soumana Hima

Authors
  1. Hanwu Li
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  2. Shige Peng
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  3. Abdoulaye Soumana Hima
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Correspondence to Shige Peng.

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Li, H., Peng, S. & Soumana Hima, A. Reflected solutions of backward stochastic differential equations driven by G-Brownian motion. Sci. China Math. 61, 1–26 (2018). https://doi.org/10.1007/s11425-017-9176-0

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  • DOI: https://doi.org/10.1007/s11425-017-9176-0

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