Skip to main content

Convergence at First and Second Order of Some Approximations of Stochastic Integrals

Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 2006)

Abstract

We consider the convergence of the approximation schemes related to Itô’s integral and quadratic variation, which have been developed in Russo and Vallois (Elements of stochastic calculus via regularisation, vol. 1899, pp. 147–185, Springer, Berlin, 2007). First, we prove that the convergence in the a.s. sense exists when the integrand is Hölder continuous and the integrator is a continuous semimartingale. Second, we investigate the second order convergence in the Brownian motion case.

  • Stochastic integration by regularization
  • Quadratic variation
  • First and second order convergence
  • Stochastic Fubini’s theorem

2000 MSC: 60F05, 60F17, 60G44, 60H05, 60J65

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bérard Bergery, B.: Approximation du temps local et intégration par régularisation. PhD thesis, Nancy Université (2004–2007)

    Google Scholar 

  2. Gradinaru, M., Nourdin, I.: Approximation at first and second order of m-order integrals of the fractional Brownian motion and of certain semimartingales. Electron. J. Probab., 8(18), 26 pp. (electronic) (2003)

    Google Scholar 

  3. Jacod, J., Mémin, J.: Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité. In Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980), Lecture Notes in Mathematics, vol. 850, pp. 529–546. Springer (1981)

    Google Scholar 

  4. Jakubowski, A., Mémin, J., Pagès, G.: Convergence en loi des suites d’intégrales stochastiques sur l’espace D 1 de Skorokhod. Probab. Theor. Relat. Field. 81(1), 111–137 (1989)

    CrossRef  MATH  Google Scholar 

  5. Karandikar, R.: On almost sure convergence results in stochastic calculus. In Séminaire de Probabilités, XXXIX, Lecture Notes in Mathematics, vol. 1874, pp. 137–147, Springer, Berlin (2006)

    Google Scholar 

  6. Meyer, P.A.: Probabilités et potentiel, Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. XIV. Actualités Scientifiques et Industrielles, No. 1318, Hermann, Paris (1966)

    Google Scholar 

  7. Nualart, D., Peccati, G.: Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33(1), 177–193 (2005)

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Revuz, D., Yor, M.: Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Third edition. Springer, Berlin (1999)

    Google Scholar 

  9. Rootzen, H.: Limit distribution for the error in approximations of stochastic integrals. Ann. Probab. 8(2), 241–251 (1980)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Russo, F., Vallois, P.: Elements of stochastic calculus via regularisation. In Séminaire de Probabilités XL, Lecture Notes in Mathematics, vol. 1899, pp. 147–185, Springer, Berlin (2007)

    Google Scholar 

  11. Russo, F., Vallois, P.: Itô formula for C 1-functions of semimartingales. Probab. Theor. Relat. Field. 104(1), 27–41 (1996)

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Russo, F., Vallois, P.: Stochastic calculus with respect to continuous finite quadratic variation processes. Stochast. Rep. 70(1–2), 1–40 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Russo, F., Vallois, P.: The generalized covariation process and Itô formula. Stoch. Process. Appl. 59(1), 81–104 (1995)

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Blandine Bérard Bergery .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Bergery, B.B., Vallois, P. (2011). Convergence at First and Second Order of Some Approximations of Stochastic Integrals. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLIII. Lecture Notes in Mathematics(), vol 2006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15217-7_10

Download citation