Advertisement

Potential Analysis

, Volume 26, Issue 4, pp 363–396 | Cite as

Kusuoka–Stroock Formula on Configuration Space and Regularities of Local Times with Jumps

  • Jiagang Ren
  • Michael RöcknerEmail author
  • Xicheng Zhang
Article

Abstract

In this paper, we first extend the classical Itô stochastic integral to the case of measurable fields of Hilbert spaces. Then, a Kusuoka–Stroock formula on configuration space is proved. Using this formula, we study the fractional regularities of local times with jumps in the sense of the Malliavin calculus.

Keywords

Hilbert measurable fields Stochastic integral Kusuoka–Stroock formula Configuration space Local time 

Mathematics Subject Classifications (2000)

60H05 60H07 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Airault, H., Ren, J., Zhang, X.: Smoothness of local times of semimartingales. C. R. Acad. Sci. Paris Sér. I. Math. 330, 719–724 (2000)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Albeverio, S., Kontratiev, Yu. G., Röckner, M.: Analysis and geometry on configuration spaces. J. Funct. Anal. 154, 444–500 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Aubin, T.: Nonlinear Analysis on Manifolds. Monge-Ampère Equations. Springer, Berlin Heidelberg New York (1982)zbMATHGoogle Scholar
  4. 4.
    Bouleau, N., Hirsch, F.: Dirichlet forms and analysis on Wiener space. de Gruyter Studies in Mathematics, vol. 14. Walter de Gruyter, Berlin (1991)Google Scholar
  5. 5.
    Bicheteler, K.: Stochastic integration with jumps. Encyclopedia Math. Appl. 89 (2002)Google Scholar
  6. 6.
    Bichteler, K., Gravereaux, J.B., Jacod, J.: Malliavin calculus for processes with jumps. In: Stochastic Graph, vol. 2. Gordon and Breach, New York (1987)Google Scholar
  7. 7.
    Bogachev, V.I., Pugachev, O.V., Röckner, M.: Surface measure and tightness of (r, p)-capacity on Poisson space. J. Funct. Anal. 196, 61–86 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cruzeiro, A.B., Zhang, X.: L p-gradient estimates of symmetric Markov semigroups for 1 < p⩽2. Acta Math. Sinica 22(1), 101–104 (2002)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  10. 10.
    He, S., Wang, J., Yan, J.: Semimartingale Theory and Stochastic Calculus. C.R.C. Science (1992)Google Scholar
  11. 11.
    Hu, J., Ren, J.: Infinite dimensional quasi continuity, path continuity and ray continuity of functions with fractional regularity. J. Math. Pures Appl. 80(1), (2001)Google Scholar
  12. 12.
    Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes (2nd edition). North-Holland Mathematical Library, vol. 24. North-Holland, Amsterdam, The Netherlands (1989)Google Scholar
  13. 13.
    Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus. I. Stochastic Analysis (Katata/Kyoto, 1982), vol. 32, pp. 271–306. North-Holland Math. Library, North-Holland, Amsterdam, The Netherlands (1984)Google Scholar
  14. 14.
    Ma, Z.-M., Röckner, M.: An Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Springer, Berlin Heidelberg New York (1992)Google Scholar
  15. 15.
    Ma, Z., Röckner, M.: Construction of diffusions on configuration spaces. Osaka J. Math. 56, 273–314 (2000)Google Scholar
  16. 16.
    Malliavin, P.: Stochastic analysis. Grundlehren der Mathematischen Wissenschaften, vol. 313. Springer, Berlin Heidelberg New York (1997)Google Scholar
  17. 17.
    Nualart, D., Vives, J.: Smoothness of local time and related Wiener functionals. Potential Anal. 1, 257–263 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Ren, J., Zhang, X.: Path continuity of fractional Dirichlet functionals. Bull. Sci. Math. 127(4), 368–378 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian motion Grundlehren der Mathematischen Wissenschaften, vol. 293. 2nd ed. Springer, Berlin Heidelberg New York (1994)Google Scholar
  20. 20.
    Shigekawa, I.: On a quasi-everywhere existence of the local time of the 1-dimensional Brownian motion. Osaka J. Math. 21(3), 621–627 (1984)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Takesaki, M.: Theory of Operator Algebra I. Springer, Berlin Heidelberg New York (1979)Google Scholar
  22. 22.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam, The Netherlands (1978)Google Scholar
  23. 23.
    Watanabe, S.: Wiener functionals with the regularity of fractional order, in new trends in stochastic analysis. In: Elworthy, K.D., Kusuoka, S., Shigekawa, I. (eds.) World Scientific, Singapore (1997)Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  • Jiagang Ren
    • 1
  • Michael Röckner
    • 2
    • 4
    Email author
  • Xicheng Zhang
    • 3
    • 4
  1. 1.School of Mathematics and Computational ScienceZhongshan UniversityGuangzhouPeople’s Republic of China
  2. 2.Departments of Mathematics and StatisticsPurdue UniversityW. LafayetteUSA
  3. 3.Department of MathematicsHuazhong University of Science and TechnologyWuhanPeople’s Republic of China
  4. 4.Fakultät für MathematikUniversität BielefeldBielefeldGermany

Personalised recommendations