Advertisement

Journal of Theoretical Probability

, Volume 13, Issue 1, pp 193–224 | Cite as

Quadratic Covariation and Itô's Formula for Smooth Nondegenerate Martingales

  • S. Moret
  • D. Nualart
Article

Abstract

In this paper we prove the existence of the quadratic covariation [f(X),X], where f is a locally square integrable function and Xt = ∫t0usdWs is a smooth nondegenerate Brownian martingale. This result is based on some moment estimates for Riemann sums which are established by means of the techniques of the Malliavin calculus.

Itô's formula Malliavin calculus quadratic covariation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    Bardina, X., and Jolis, M. (1997). An extension of Itô's formula for elliptic diffusion processes. Stoch. Proc. Appl. 69, 83–109.CrossRefGoogle Scholar
  2. 2.
    Bouleau, N., and Yor, M. (1981). Sur la variation quadratique des temps locaux de certaines semimartingales. C. R. Acad. Sci. Paris 292, 491–494.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bouleau, N., and Hirsh, F. (1986). Propriétés d'absolute continuité dans les espaces de Dirichlet et applications aux équations différentielles stochastiques. In Séminaire de Probabilités XX, Lecture Notes in Math. 1204, 131–161.Google Scholar
  4. 4.
    Eisenbaum, N. (1997). Integration with respect to local time. Preprint.Google Scholar
  5. 5.
    Föllmer, H., Protter, Ph., and Shiryayev, A. N. (1995). Quadratic covariation and an extension of Itoô's formula. Bernouilli 1 (1–2), 149–169.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Föllmer, H., and Protter, Ph. (1997). On Itoô's formula for d-dimensional Brownian motion. Preprint.Google Scholar
  7. 7.
    Itoô, K. (1944). Stochastic integral. Proc. Imperial Acad. Tokyo 20, 519–524.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Nualart, D. (1998). Analysis on Wiener space and anticipating stochastic calculus. In: École d'été de Saint-Flour XXV (1995). Lect. Notes in Math. 1690, 123–227.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Nualart, D. (1995). Malliavin Calculus and Related Topics, Springer-Verlag.Google Scholar
  10. 10.
    Russo, F., and Vallois, P. (1994). Itoô's formula for C1 functions of semimartingales. Prob. Th. Rel. Fields 104 (1), 27–41.CrossRefGoogle Scholar
  11. 11.
    Wolf, J. (1997a). An Itoô's formula for local Dirichlet processes. Stoch. Stoch. Rep. 62, 103–115.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Wolf, J. (1997b). Transformations on semimartingales and local Dirichlet processes. Stoch. Stoch. Rep. 62, 65–101.MathSciNetCrossRefGoogle Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • S. Moret
  • D. Nualart

There are no affiliations available

Personalised recommendations