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
General change of variable formulas for semimartingales in one and finite dimensions
Download PDF
Download PDF
  • Published: September 1993

General change of variable formulas for semimartingales in one and finite dimensions

  • Philip Protter1 &
  • Jaime San Martin2 

Probability Theory and Related Fields volume 97, pages 363–381 (1993)Cite this article

  • 201 Accesses

  • 4 Citations

  • Metrics details

Summary

A general one dimensional change of variables formula is established for continuous semimartingales which extends the famous Meyer-Tanaka formula. The inspiration comes from an application arising in stochastic finance theory. For functions mapping ℝn to ℝ, a general change of variables formula is established for arbitrary semimartingales, where the usualC 2 hypothesis is relaxed.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Brosamler, G.: Quadratic variation of potentials and harmonic functions. Trans. Am. Math. Soc.149, 243–257 (1970)

    Google Scholar 

  2. Carlen, E., Protter, P.: On semimartingale decompositions of convex functions of semimartingales. Ill. J. Math.36, 420–427 (1992)

    Google Scholar 

  3. Çinlar, E., Jacod, J., Protter, P., Sharpe, M.: Semimartingales and Markov processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.54, 161–220 (1980)

    Google Scholar 

  4. Kunita, H.: Some extensions of Itô's formula. In: Azéma, J., Yor, M. (eds.) Semin. de Probab. XV. (Lect. Notes Math., vol. 850, pp. 118–141) Berlin Heidelberg New York: Springer 1981

    Google Scholar 

  5. Krylov, N. V.: Controlled diffusion processes. Berlin Heidelberg New York: Springer 1980

    Google Scholar 

  6. Kubo, I.: Itô formula for generalized Brownian functionals. Lect. Notes Control Inf. Sci.49, 156–166 (1983)

    Google Scholar 

  7. Kuo, H. H., Shieh, N. R.: A generalized Itô's formula for multidimensional Brownian motion and its applications. Chinese J. Math.15, 163–174 (1987)

    Google Scholar 

  8. Meyer, P. A.: Un cours sur les intégrales stochastiques. In: Meyer, P. A. (ed.) Sémin. Probab. X. (Lect. Notes Math., vol. 511, pp. 246–400) Berlin Heidelberg New York: Springer 1976

    Google Scholar 

  9. Meyer, P. A.: La formule d'Itô pour le mouvement Brownien d'après G. Brosamler. In: Dellocherie, C. et al (eds.) Sémin. Probab. XII. (Lect. Notes Math., vol. 649, pp. 763–769) Berlin Heidelberg New York: Springer 1978

    Google Scholar 

  10. Myneni, R.: Personal communications (1990, 1991)

  11. Protter, P.: Stochastic integration and differential equations. A new approach. Berlin Heidelberg New York: Springer 1990

    Google Scholar 

  12. Ouknine, Y., Rutkowski, M.: Local times of functions of continuous semimartingales. (Preprint)

  13. Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Berlin Heidelberg New York: Springer 1990

    Google Scholar 

  14. Rosen, J.: A representation for the intersection local time of Brownian motion in space. Ann. Probab.13, 145–153 (1985)

    Google Scholar 

  15. San Martin, J.: Stochastic differential equations. Ph.D. Thesis, Purdue University 1990

  16. San Martin, J.: One dimensional stochastic differential equations. Ann. Probab.21, 509–553 (1993)

    Google Scholar 

  17. Sznitman, A. S., Varadhan, S. R. S.: A multidimensional process involving local time. Probab. Theory Relat. Fields71, 553–579 (1986)

    Google Scholar 

  18. Yor, M.: Compléments aux formules de Tanaka-Rosen. In: Azéma, J., Yor, M. (eds.) Sémin. de Probab. XIX. (Lect. Notes Math., vol. 1123, pp. 332–349) Berlin Heidelberg New York: Springer 1985

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics and Statistics Departments, Purdue University, 47907-1395, W. Lafayette, IN, USA

    Philip Protter

  2. Facultad de Ciencias Fis.y Mat., Depto. Ingeniería Matematica, Universidad de Chile, Casilla 170/3, Santiago, Chile

    Jaime San Martin

Authors
  1. Philip Protter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jaime San Martin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported in part by NSF grant No. DMS-9103454

Supported in part by John D. and Catherine T. MacArthur Foundation award for US-Chile Scientific Cooperation

Supported in part by FONDECYT, grant 92-0881

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Protter, P., San Martin, J. General change of variable formulas for semimartingales in one and finite dimensions. Probab. Th. Rel. Fields 97, 363–381 (1993). https://doi.org/10.1007/BF01195071

Download citation

  • Received: 18 November 1991

  • Revised: 25 May 1993

  • Issue Date: September 1993

  • DOI: https://doi.org/10.1007/BF01195071

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

Mathematics Subject Classification

  • 60H05
  • 60G44
  • 60G07
  • 60H10
  • 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