Skip to main content

Advertisement

SpringerLink
Log in

Maximal Inequalities of the Itô Integral with Respect to Poisson Random Measures or Lévy Processes on Banach Spaces

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

We are interested in maximal inequalities satisfied by a stochastic integral driven by a Poisson random measure in a general Banach space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. de Acosta, A., Araujo, A., Giné, E.: On Poisson measures, Gaussian measures and the central limit theorem in Banach spaces. In: Probability on Banach Spaces. Adv. Probab. Relat. Top., vol. 4, pp. 1–68. Dekker, New York (1978)

    Google Scholar 

  2. Albeverio, S., Rüdiger, B.: Stochastic integrals and the Lévy–Ito decomposition theorem on separable Banach spaces. Stoch. Anal. Appl. 23, 217–253 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Applebaum, D.: Lévy processes and stochastic calculus. In: Cambridge Studies in Advanced Mathematics, vol. 93. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  4. Applebaum, D.: Lévy processes and stochastic integrals in Banach spaces. Probab. Math. Stat. 27, 75–88 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Araujo, A., Giné, E.: Type, cotype and Lévy measures in Banach spaces. Ann. Probab. 6, 637–643 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bass, R.F., Cranston, M.: The Malliavin calculus for pure jump processes and applications to local time. Ann. Probab. 14, 490–532 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brzeźniak, Z.: Stochastic partial differential equations in M-type 2 Banach spaces. Potential Anal. 4, 1–45 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brzeźniak, Z., Hausenblas, E.: Maximal regularity for stochastic convolutions driven by Lévy processes. Probab. Theory Relat. Fields 145, 615–637 (2009)

    Article  MATH  Google Scholar 

  9. Brzeźniak, Z., Hausenblas, E.: Martingale Solutions for Stochastic Equation of Reaction Diffusion Type Driven by Lévy Noise or Poisson Random Measure (2010, submitted)

  10. Burkholder, D.: Distribution function inequalities for martingales. Ann. Probab. 1, 19–42 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  11. Burkholder, D., Davis, B., Gundy, R.: Integral inequalities for convex functions of operators on martingales. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. II, pp. 223–240. University of California Press, Berkeley (1972)

    Google Scholar 

  12. Davis, B.: On the integrability of the Martingale square function. Isr. J. Math. 8, 187–190 (1970)

    Article  MATH  Google Scholar 

  13. Dettweiler, E.: Banach space valued processes with independent increments and stochastic integration. In: Probability in Banach spaces, vol. IV (Oberwolfach, 1982). Lecture Notes in Mathematics, vol. 990, pp. 54–83. Springer, Berlin (1983)

    Chapter  Google Scholar 

  14. Dettweiler, E.: A characterization of the Banach spaces of type p by Levy measures. Math. Z. 157, 121–130 (1077)

    Article  MathSciNet  Google Scholar 

  15. Garsia, A.: On a convex function inequality for martingales. Ann. Probab. 1, 171–174 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland Mathematical Library, vol. 24. North-Holland, Amsterdam (1989)

    MATH  Google Scholar 

  17. Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 288. Springer, Berlin (2003)

    MATH  Google Scholar 

  18. Kallenberg, O.: Foundations of modern probability. In: Probability and its Applications (New York), 2nd edn. Springer, New York (2002)

    Google Scholar 

  19. Krasnoselśkiī, M., Rutickiī, J.: Convex Functions and Orlicz Spaces. Translated from the First Russian Edition by Leo F. Boron. Noordhoff, Groningen (1961)

  20. Linde, W.: Probability in Banach Spaces—Stable and Infinitely Divisible Distributions, 2nd edn. Wiley-Interscience, New York (1986)

    MATH  Google Scholar 

  21. Marinelli, C., Prévôt, C., Röckner, M.: Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise (English summary). J. Funct. Anal. 258(2), 616–649 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. McConnell, T.: Decoupling and stochastic integration in UMD Banach spaces. Probab. Math. Stat. 10, 283–295 (1989)

    MathSciNet  MATH  Google Scholar 

  23. Neidhardt, A.L.: Stochastic Integrals in 2-Uniformly Smooth Banach Spaces. Ph.D. thesis, University of Wisconsin (1978)

  24. Riedle, M., van Gaans, O.: Stochastic integration for Lévy processes with values in Banach spaces. Stoch. Process. Their Appl. 119, 1952–1974 (2009)

    Article  MATH  Google Scholar 

  25. Peszat, S., Zabczyk, J.: Stochastic partial differential equations with Lévy noise. In: Encyclopedia of Mathematics and its Applications, vol. 113. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  26. Protter, P., Talay, D.: The Euler scheme for Lévy driven stochastic differential equations. Ann. Probab. 25, 393–423 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Roberts, W., Varberg, D.: Convex functions. In: Pure and Applied Mathematics, vol. 57. Academic, New York (1973)

    Google Scholar 

  28. Rüdiger, B.: Stochastic integration with respect to compensated Poisson random measures on separable Banach spaces. Stoch. Stoch. Rep. 76, 213–242 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. van Neerven, J., Veraar, M., Weis, L.: Stochastic integration in UMD Banach spaces. Ann. Probab. 35, 1438–1478 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Information Technology, Montana University of Leoben, Franz-Josef-Straße 18, Leoben, Austria

    Erika Hausenblas

Authors
  1. Erika Hausenblas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Erika Hausenblas.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hausenblas, E. Maximal Inequalities of the Itô Integral with Respect to Poisson Random Measures or Lévy Processes on Banach Spaces. Potential Anal 35, 223–251 (2011). https://doi.org/10.1007/s11118-010-9210-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-010-9210-0

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation