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On the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with integral-Lipschitz coefficients

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Abstract

In this paper, we study the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion (GSDEs) with integral-Lipschitz coefficients.

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References

  1. Bihari, I. A generalization of a lemma of bellman and its application to uniqueness problems of differential equations. Acta Math. Acad. Sci. Hungar., 7: 81–94 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  2. Denis, L., Hu, M., Peng, S. Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths. Potential Anal., 34(2): 139–161 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Fang, S., Zhang, T. A study of a class of stochastic differential equations with non-Lipschitzian coefficients. Probab. Theory Related Fields, 132(3): 356–390 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Gao, F. Pathwise properties and homomorphic flows for stochastic differential equations driven by G-Brownian motion. Stochastic Process. Appl., 119(10): 3356–3382 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hu, M., Ji, S., Peng, S., Song, Y. Backward stochastic differential equations driven by G-Brownian motion. Stochastic Process. Appl., 124(1): 759–784 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  6. Hu, Y., Lerner, N. On the existence and uniqueness of solutions to stochastic equations in infinite dimension with integral-Lipschitz coefficients. J. Math. Kyoto Univ., 42(3): 579–598 (2002)

    MATH  MathSciNet  Google Scholar 

  7. Li, X., Peng, S. Stopping times and related Itô’s calculus with G-Brownian motion. Stochastic Process. Appl., 121(7): 1492–1508 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Lin, Q. Some properties of stochastic differential equations driven by the G-Brownian motion. Acta Math. Appl. Sinica (English Ser.), 29(5): 923–942 (2013)

    Article  MATH  Google Scholar 

  9. Peng, S. G-expectation, G-Brownian motion and related stochastic calculus of Itô type. In: Stochastic analysis and applications, Abel Symp., Vol.2, ed. by F.E., Benth, G. Di Nunno, T. Lindstrom, B. Øksendal, T. Zhang, Springer-Verlag, Berlin, 2007, 541–567

    Chapter  Google Scholar 

  10. Peng, S. Nonlinear expectations and stochastic calculus under uncertainty. arXiv:1002.4546v1

  11. Peng, S., Song, Y., Zhang, J. A complete representation theorem for G-martingales. arXiv:1201.2629v2

  12. Rockafellar, R. T. Convex analysis. Princeton University Press, Princeton, N.J., 1970

    MATH  Google Scholar 

  13. Soner, H. M., Touzi, N., Zhang, J. Martingale representation theorem under G-expectation. Stochastic Process. Appl., 121(2): 265–287 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Song, Y. Some properties on G-evaluation and its applications to G-martingale decomposition. Sci. China Math., 54(2): 287–300 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Song, Y. Uniqueness of the representation for G-martingales with finite variation. Electron. J. Probab., 17(24): 1–15 (2012)

    MathSciNet  Google Scholar 

  16. Watanabe, S., Yamada, T. On the uniqueness of solutions of stochastic differential equations II. J. Math. Kyoto Univ., 11: 553–563 (1971)

    MATH  MathSciNet  Google Scholar 

  17. Xu, J., Zhang, B. Martingale characterization of G-Brownian motion. Stochastic Process. Appl., 119(1): 232–248 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Yamada, T. On the successive approximation of solutions of stochastic differential equations. J. Math. Kyoto Univ., 21(3): 501–515 (1981)

    MATH  MathSciNet  Google Scholar 

  19. Yamada, T., Watanabe, S. On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ., 11: 155–167 (1971)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institut de Recherche Mathématique de Rennes, Université de Rennes 1, 35042, Rennes Cedex, France

    Xue-peng Bai & Yi-qing Lin

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

    Xue-peng Bai & Yi-qing Lin

Authors
  1. Xue-peng Bai
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  2. Yi-qing Lin
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Corresponding author

Correspondence to Yi-qing Lin.

Additional information

The research is partially supported by the Major Program in Key Research Institute of Humanities and Social Sciences sponsored by Ministry of Education of China (under grant No. 2009JJD790049) and the Post-graduate Study Abroad Program sponsored by China Scholarship Council.

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Bai, Xp., Lin, Yq. On the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with integral-Lipschitz coefficients. Acta Math. Appl. Sin. Engl. Ser. 30, 589–610 (2014). https://doi.org/10.1007/s10255-014-0405-9

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  • DOI: https://doi.org/10.1007/s10255-014-0405-9

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