Skip to main content

Some remarks on the theory of stochastic integration

Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 1485)

Abstract

The main purpose of this article is to propose a reasonable definition for the stochastic integration (S.I.) of progressive processes w.r.t. semimartingales. This S.I. generalizes that of predictable processes w.r.t. semimartingales as well as the stochastic Stieltjes integration. This S.I. is proposed in §1. We give also in §1 an exponential formula for semimartingales using this S.I.. The rest of this paper consists of several remarks on the theory of stochastic integration which are mostly of pedagogical interest. In §2 we propose a new construction of the S.I. of predictable processes w.r.t. local martingales. A simple proof of the integration by parts formula is given in §3. Finally, we propose in §4 a short proof of Meyer’s theorem on compensated stochastic integrals of local martingales.

AMS Subject Classification

  • 60H05

The project supported by the National Natural Science Foundation of China.

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   44.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.00
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. C. Dellacherie, P.A. Meyer: Probabilités et potenliel, vol II, Hermann, Pairs, 1982.

    Google Scholar 

  2. J. Jacod, A. N. Shiryaev: Lemit Theorems for Stochastic Processes, Springer-Verlag, 1987.

    Google Scholar 

  3. J. Jacod: Sur la construction des intégrales stochastiques et les sous-espaces stables des martingales, Sém. Probab. XI, LN in Math. 581, 1977.

    Google Scholar 

  4. P.A.Meyer: Un cours sur les intégrales stochastiques, Sém. Probab. X, LN in Math. 511, 1976.

    Google Scholar 

  5. J.A. Yan: Remarques sur l’intégrale stochastique de processus non bornés, Sém. Probab. XIV, LN in Math. 784, 1980.

    Google Scholar 

  6. J.A. Yan: Remarques sur certaines classes des semimartingales et sur les intégrales slochasliques optionnelles, ibid.

    Google Scholar 

  7. M. Yor: En cherchant une définition naturelle des intégrales stochastiques optionnelles, Sém. Probab. XIII, LN in Math. 721, 1979.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1991 Springer-Verlag

About this paper

Cite this paper

Yan, J.A. (1991). Some remarks on the theory of stochastic integration. In: Azéma, J., Yor, M., Meyer, P.A. (eds) Séminaire de Probabilités XXV. Lecture Notes in Mathematics, vol 1485. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0100849

Download citation

  • DOI: https://doi.org/10.1007/BFb0100849

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54616-0

  • Online ISBN: 978-3-540-38496-0

  • eBook Packages: Springer Book Archive