Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
The chaotic-representation property for a class of normal martingales
Download PDF
Download PDF
  • Published: 20 January 2007

The chaotic-representation property for a class of normal martingales

  • Stéphane Attal1 &
  • Alexander C. R. Belton2 

Probability Theory and Related Fields volume 139, pages 543–562 (2007)Cite this article

  • 125 Accesses

  • 7 Citations

  • Metrics details

Abstract

Suppose \(Z=(Z_t)_{t\ge0}\) is a normal martingale which satisfies the structure equation

$$d[Z]_t = (\alpha(t)+\beta(t)Z_{t-}) dZ_t + dt$$

. By adapting and extending techniques due to Parthasarathy and to Kurtz, it is shown that, if α is locally bounded and β has values in the interval [-2,0], the process Z is unique in law, possesses the chaotic-representation property and is strongly Markovian (in an appropriate sense). If also β is bounded away from the endpoints 0 and 2 on every compact subinterval of [0,∞] then Z is shown to have locally bounded trajectories, a variation on a result of Russo and Vallois.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Amendinger J. (2000). Martingale representation theorems for initially enlarged filtrations. Stoch. Process. Appl. 89: 101–116

    Article  MATH  Google Scholar 

  2. Belton, A.C.R.: The monotone Poisson process. In: Proceedings of the 25th Conference on Quantum Probability and Related Topics, Bȩdlewo, 2004. Banach Center Publications, Warsaw (to appear)(2006)

  3. Belton, A.C.R.: On the path structure of a particular Azéma martingale. University College, Cork (Preprint)(2006)

  4. Brémaud P., Yor M. (1978). Changes of filtrations and of probability measures. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 45: 269–295

    Article  MATH  Google Scholar 

  5. Dellacherie C., Maisonneuve B., Meyer P.-A. (1992). Probabilités et potentiel, Chapitres XVII–XXIV: processus de Markov (fin), compléments de calcul stochastique. Hermann, Paris

    Google Scholar 

  6. Dermoune A. (1990). Distributions sur l’espace de P. Lévy et calcul stochastique. Ann. Inst. Henri Poincaré, Probab. Stat. 26: 101–119

    MATH  Google Scholar 

  7. Dritschel M., Protter P. (1999). Complete markets with discontinuous security price. Finance Stoch. 3: 203–214

    Article  MATH  Google Scholar 

  8. Émery M. (1989). On the Azéma martingales. In: Azéma, J., Meyer, P.-A., Yor, M. (eds) Sémin, Probab, XXIII, Lecture Notes on Mathematics, vol 1372, pp 66–87. Springer, Heidelberg

    Google Scholar 

  9. Émery M. (1996). On the chaotic representation property for martingales. In: Ibragimov, I.A., Zaitsev, A.Yu. (eds) Probability Theory and Mathematical Statistics, St. Petersburg, 1993, pp 155–166. Gordon and Breach, Amsterdam

    Google Scholar 

  10. Hudson R.L., Parthasarathy K.R. (1984). Quantum Ito’s formula and stochastic evolutions. Commun. Math. Phys. 93: 301–323

    Article  MATH  Google Scholar 

  11. Kurtz D. (2003). Répresentation nucléaire des martingales d’Azéma. In: Azéma, J., Émery, M., Ledoux, M., Yor, M. (eds) Sémin. Probab. XXXVI. Lecture Notes on Mathematics, vol. 1801, pp 457–476. Springer, Heidelberg

    Google Scholar 

  12. Le Jan Y. (1979). Martingales et changement de temps. In: Dellacherie, C., Meyer, P.-A., Weil, M. (eds) Sémin. Probab. XIII. Lecture Notes on Mathematics, vol. 721, pp 385–399. Springer, Heidelberg

    Google Scholar 

  13. Parthasarathy, K.R. (1990) Azéma martingales and quantum stochastic calculus. In: Bahadur, R.R. (ed.) Proceedings of the R.C. Bose Symposium on Probability, Statistics and Design of Experiments, pp. 551–569. Wiley Eastern, New Delhi (1990)

  14. Parthasarathy K.R. (1995). Azéma martingales with drift. Probab. Math. Stat. 15: 461–468

    MATH  Google Scholar 

  15. Privault N., Solé J.L., Vives J. (2000). Chaotic Kabanov formula for the Azéma martingales. Bernoulli 6: 633–651

    Article  MATH  Google Scholar 

  16. Protter P. (1990). Stochastic integration and differential equations. A new approach. Applications of Mathematics, vol. 21. Springer, Heidelberg

    Google Scholar 

  17. Reed M., Simon B. (1972). Methods of modern mathematical physics. I: Functional analysis. Academic, New York

    Google Scholar 

  18. Reed M., Simon B. (1975). Methods of modern mathematical physics. II: Fourier analysis, self-adjointness. Academic, New York

    MATH  Google Scholar 

  19. Rogers L.C.G., Williams D. (2000). Diffusions, Markov processes and martingales, vol 1 Foundations. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  20. Russo F., Vallois P. (1998). Product of two multiple stochastic integrals with respect to a normal martingale. Stoch. Process. Appl. 73: 47–68

    Article  MATH  Google Scholar 

  21. Taviot, G.: Martingales et équations de structure : étude géométrique. Thèse, Université Louis Pasteur Strasbourg 1 (1999)

  22. Utzet F. (1992). Les processus à accroissements indépendants et les équations de structure. In: Azéma, J., Meyer, P.-A., Yor, M. (eds) Sémin, Probab, XXVI, Lecture Notes on Mathematics, vol 1526, pp 405–409. Springer, Heidelberg

    Google Scholar 

  23. Yor, M.: Sous-espaces denses dans L 1 ou H 1 et représentation des martingales (avec J. de Sam Lazaro pour l’appendice). In: Dellacherie, C., Meyer, P.-A., Weil, M. (eds.) Sémin. Probab. XII. Lecture Notes on Mathematics, vol. 649, pp. 265–309 (1978)

Download references

Author information

Authors and Affiliations

  1. Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 avenue du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Stéphane Attal

  2. School of Mathematics, Applied Mathematics and Statistics, University College, Cork, Ireland

    Alexander C. R. Belton

Authors
  1. Stéphane Attal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander C. R. Belton
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander C. R. Belton.

Additional information

A. C. R. Belton acknowledges support from the European Community’s Human Potential Programme under contract HPRN-CT-2002-00279, QP-Applications.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Attal, S., Belton, A.C.R. The chaotic-representation property for a class of normal martingales. Probab. Theory Relat. Fields 139, 543–562 (2007). https://doi.org/10.1007/s00440-006-0052-z

Download citation

  • Received: 05 June 2006

  • Revised: 14 November 2006

  • Published: 20 January 2007

  • Issue Date: November 2007

  • DOI: https://doi.org/10.1007/s00440-006-0052-z

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Azéma martingale
  • Chaotic-representation property
  • Normal martingale
  • Predictable-representation property
  • Structure equation

Mathematics Subject Classification (2000)

  • 60G44
  • 60H20
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature