Skip to main content
SpringerLink
Log in

Multiplicative Decompositions and Frequency of Vanishing of Nonnegative Submartingales

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

In this paper, we establish a multiplicative decomposition formula for nonnegative local martingales and use it to characterize the set of continuous local submartingales Y of the form Y = N + A, where the measure dA is carried by the set of zeros of Y. In particular, we shall see that in the set of all local submartingales with the same martingale part in the multiplicative decomposition, these submartingales are the smallest ones. We also study some integrability questions in the multiplicative decomposition and interpret the notion of saturated sets in the light of our results.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Azéma J. (1972). Quelques applications de la théorie générale des processus I. Invent. Math. 18, 293–336

    Article  MathSciNet  Google Scholar 

  2. Azéma, J. (1978). Représentation d’une surmartingale bornée, ZW, 45, 191–212.

    Google Scholar 

  3. Azéma, J., Meyer, P.A., and Yor, M. (1992). Martingales relatives. Sém.Proba. XXVI, Lecture Notes in Mathematics Vol. 1526, Springer, pp.307–321.

  4. Azéma, J., and Yor, M. (1992). Sur les zéros des martingales continues, Sém.Proba. XXVI, Lecture Notes in Mathematics Vol. 1526, Springer, pp.248–306.

  5. Barlow, M.T., Emery, M., Knight, F.B., Song, S., and Yor, M. (1998). Autour d’un théorème de Tsirelson sur des filtrations browniennes et non browniennes, Sém.Proba. XXXII, Lecture Notes in Mathematics Vol. 1686, Springer, pp.264–305.

  6. Dellacherie, C., Maisonneuve, B., and Meyer, P.A. (1992). Probabilités et Potentiel, Chapitres XVII-XXIV: Processus de Markov (fin), Compléments de Calcul Stochastique, Hermann Paris.

  7. Itô K., Watanabe S. (1965). Transformation of Markov processes by multiplicative functionals. Ann. Inst. Fourier. Grenoble. 15(1): 13–30

    MathSciNet  Google Scholar 

  8. Jacod, J. (1979).Calcul stochastique et problèmes de martingales, Lecture Notes in Mathematics Vol. 714, Springer pp. 240–249.

  9. Meyer, P.A. (1979). Représentations multiplicatives de sousmartingales d’après Azéma, Sém.Proba. XIII, Lecture Notes in Mathematics, Vol. 721, Springer.

  10. Meyer, P.A. and Yoeurp, C. (1979). Sur la décomposition multiplicative des sousmartingaless positives, Sém.Proba. XIII, Lecture Notes in Mathematics, Vol. 721, Springer, pp.240–249.

  11. Nikeghbali A., Yor M. (2006). Doob’s maximal identity, multiplicative decompositions and enlargements of filtrations. Illinois J Math. 50(4): 791–814

    MathSciNet  Google Scholar 

  12. Nikeghbali A. (2006). A class of remarkable martingales, Stochastic Process. Appl. 116(6): 917–938

    MathSciNet  Google Scholar 

  13. Nikeghbali A. (2006). Enlargements of filtrations and path decompositions at non stopping times. Probab. Theory and Relat. Fields 136, 532–540

    Google Scholar 

  14. Revuz D., Yor M. (1999). Continuous martingales and Brownian Motion, 3rd ed. Springer, Berlin

    MATH  Google Scholar 

  15. Yor M. (1979). Les inégalités de sous-martingales comme conséquence de la relation de domination. Stochastics 3,1–15

    MathSciNet  Google Scholar 

  16. Yor M. (1997). Some Aspects of Brownian Motion, Part II. Some Recent Martingale Problems. Birkhauser, Basel

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departement Mathematik, ETH Zurich, HG G16, Ramistrasse 101, 8092, Zurich, Switzerland

    Ashkan Nikeghbali

Authors
  1. Ashkan Nikeghbali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ashkan Nikeghbali.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nikeghbali, A. Multiplicative Decompositions and Frequency of Vanishing of Nonnegative Submartingales. J Theor Probab 19, 931–949 (2006). https://doi.org/10.1007/s10959-006-0035-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-006-0035-0

Keywords

2000 Mathematics Subject Classification

Search

Navigation