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G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion

  • Xi Geng
  • Zhongmin Qian
  • Danyu Yang
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2123)

Abstract

The present article is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the viewpoint of rough path theory. As the starting point, by using techniques in rough path theory, we show that quasi-surely, sample paths of G-Brownian motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 < p < 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven by G-Brownian motion in the pathwise sense under the general framework of rough paths. Next we establish the fundamental relation between SDEs and RDEs driven by G-Brownian motion. As an application, we introduce the notion of SDEs on a differentiable manifold driven by G-Brownian motion and construct solutions from the RDE point of view by using pathwise localization technique. This is the starting point of developing G-Brownian motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin. The last part of this article is devoted to such construction for a wide and interesting class of G-functions whose invariant group is the orthogonal group. In particular, we establish the generating nonlinear heat equation for such G-Brownian motion on a Riemannian manifold. We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian motion of independent interest.

Keywords

Euler-Maruyama approximation G-Brownian motion G-expectation Geometric rough paths Nonlinear diffusion processes Nonlinear heat flow Rough differential equations 

AMS classification (2010):

60H10 34A12 58J65 

Notes

Acknowledgements

The authors wish to thank Professor Shige Peng for so many valuable suggestions on the present article. The authors are supported by the Oxford-Man Institute at the University of Oxford.

References

  1. 1.
    D. Azagra, J. Ferrera, B. Sanz, Viscosity solutions to second order partial differential equations on riemannian manifolds. J. Differ. Equ. 245(2), 307–336 (2009)CrossRefMathSciNetGoogle Scholar
  2. 2.
    S. Chern, W. Chen, K. Lam, Lectures on Differential Geometry (World Scientific, Singapore, 1999)CrossRefzbMATHGoogle Scholar
  3. 3.
    G. Choquet, Theory of capacities. Ann. Inst. Fourier. 5, 87 (1953)MathSciNetGoogle Scholar
  4. 4.
    M.G. Crandall, H. Ishii, P.L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–67 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    G. de Rham, Riemannian Manifolds (Springer, New York, 1984)Google Scholar
  6. 6.
    C. Dellacherie, Capacités et Processus Stochastiques (Springer, New York, 1972)zbMATHGoogle Scholar
  7. 7.
    L. Denis, M. Hu, S. Peng, Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths. Potential Anal. 34(2), 139–161 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    K.D. Elworthy, Stochastic Differential Equations on Manifolds (Springer, New York, 1998)Google Scholar
  9. 9.
    P.K. Friz, N.B. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (Cambridge University Press, Cambridge, 2010)CrossRefGoogle Scholar
  10. 10.
    F. Gao, Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion. Stochast. Process. Appl. 119(10), 3356–3382 (2009)CrossRefzbMATHGoogle Scholar
  11. 11.
    B.M. Hambly, T.J. Lyons, Stochastic area for Brownian motion on the sierpinski gasket. Ann. Probab. 26(1), 132–148 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    E.P. Hsu, Stochastic Analysis on Manifolds (American Mathematical Society, Providence, 2002)zbMATHGoogle Scholar
  13. 13.
    N. Ikeda, S.Watanabe, Stochastic Differential Equations and Diffusion Processes (North-Holland, Amsterdam, 1989)Google Scholar
  14. 14.
    M. Kac, On distributions of certain wiener functionals. Trans. Am. Math. Soc. 65(1), 1–13 (1949)CrossRefzbMATHGoogle Scholar
  15. 15.
    I.A. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus (Springer, New York, 1991)zbMATHGoogle Scholar
  16. 16.
    S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, I and II (Interscience, New York, 1963/1969)Google Scholar
  17. 17.
    T.J. Lyons, Differential equations driven by rough signals. Rev. Mat. Iberoam. 14(2), 215–310 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    T.J. Lyons, Z. Qian, System Control and Rough Paths (Oxford University Press, Oxford, 2002)CrossRefzbMATHGoogle Scholar
  19. 19.
    T.J. Lyons, C. Michael, T. Lévy, Differential Equations Driven by Rough Paths (Springer, Berlin, 2007)zbMATHGoogle Scholar
  20. 20.
    K. Nomizu, H. Ozeki, The existence of complete riemannian metrics. Proc. Am. Math. Soc. 12(6), 889–891 (1961)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    É. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55–61 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    É. Pardoux, S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, in Stochastic Partial Differential Equations and Their Applications (Springer, New York, 1992), pp. 200–217Google Scholar
  23. 23.
    É. Pardoux, S. Peng, Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probab. Theory Relat. Fields. 98(2), 209–227 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    S. Peng, Backward SDE and related g-expectation, in Pitman Research Notes in Mathematics series, vol. 364, ed. by N. El Karoui, L. Mazliak (1997), pp. 141–159Google Scholar
  25. 25.
    S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itô type. Stochast. Anal. Appl. 2, 541–567 (2007)Google Scholar
  26. 26.
    S. Peng, Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stochast. Process. Appl. 118(12), 2223–2253 (2008)CrossRefzbMATHGoogle Scholar
  27. 27.
    S. Peng, Nonlinear expectations and stochastic calculus under uncertainty. Preprint, arXiv: 1002.4546 (2010)Google Scholar
  28. 28.
    E. Wong, M. Zakai, On the relation between ordinary and stochastic differential equations. Int. J. Eng. Sci. 3(2), 213–229 (1965)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    K. Yosida, Functional Analysis (Springer, New York, 1980)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.Oxford-Man InstituteUniversity of OxfordOxfordUK
  3. 3.Exeter CollegeUniversity of OxfordOxfordUK

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