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
Affine processes are regular
Download PDF
Download PDF
  • Published: 30 June 2010

Affine processes are regular

  • Martin Keller-Ressel1,
  • Walter Schachermayer2 &
  • Josef Teichmann1 

Probability Theory and Related Fields volume 151, pages 591–611 (2011)Cite this article

  • 327 Accesses

  • 41 Citations

  • Metrics details

Abstract

We show that stochastically continuous, time-homogeneous affine processes on the canonical state space \({\mathbb{R}_{\geq 0}^m \times \mathbb{R}^n}\) are always regular. In the paper of Duffie et al. (Ann Appl Probab 13(3):984–1053, 2003) regularity was used as a crucial basic assumption. It was left open whether this regularity condition is automatically satisfied for stochastically continuous affine processes. We now show that the regularity assumption is indeed superfluous, since regularity follows from stochastic continuity and the exponentially affine form of the characteristic function. For the proof we combine classic results on the differentiability of transformation semigroups with the method of the moving frame which has been recently found to be useful in the theory of SPDEs.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Aczél J.: Functional Equations and their Applications. Academic Press, New York (1966)

    MATH  Google Scholar 

  2. Dawson D.A., Li Z.: Skew convolution semigroups and affine Markov processes. Ann. Probab. 34(3), 1103–1142 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Duffie D., Filipovic D., Schachermayer W.: Affine processes and applications in finance. Ann. Appl. Probab. 13(3), 984–1053 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Filipović D., Teichmann J.: Regularity of finite-dimensional realizations for evolution equations. J. Funct. Anal. 197, 433–446 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Filipović, D., Tappe, S., Teichmann, J.: Jump-diffusions in Hilbert spaces: existence, stability and numerics, Stochastics (2010, forthcoming). arXiv/0810.5023

  6. Filipović, D., Tappe, S., Teichmann, J.: Term structure models driven by Wiener process and Poisson measures: existence and positivity, SIAM J. Financ. Math (2010, forthcoming). arXiv/0905.1413

  7. Jacob N.: Pseudo Differential Operators and Markov Processes, vol. I. Imperial College Press, London (2001)

    Book  Google Scholar 

  8. Kawazu K., Watanabe S.: Branching processes with immigration and related limit theorems. Theory Probab. Appl. XVI(1), 36–54 (1971)

    Article  Google Scholar 

  9. Keller-Ressel, M.: Moment explosions and long-term behavior of affine stochastic volatility models. Math. Finance (2008, forthcoming). arXiv:0802.1823

  10. Keller-Ressel, M.: Affine processes—contributions to theory and applications. PhD thesis, TU Wien (2008)

  11. Lukacs E.: Characteristic Functions. Charles Griffin & Co Ltd., London (1960)

    MATH  Google Scholar 

  12. Montgomery D., Zippin L.: Topological Transformation Groups. Interscience, New York (1955)

    MATH  Google Scholar 

  13. Rogers L.C.G., Williams D.: Diffusions, Markov Processes and Martingales, 2nd edn, vol. 1. Cambridge Mathematical Library, Cambridge (1994)

    MATH  Google Scholar 

  14. Semadeni Z.: Banach Spaces of Continuous Functions. Polish Scientific Publishers, Poland (1971)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ETH Zürich, D-Math, Rämistrasse 101, 8092, Zürich, Switzerland

    Martin Keller-Ressel & Josef Teichmann

  2. Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090, Vienna, Austria

    Walter Schachermayer

Authors
  1. Martin Keller-Ressel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Walter Schachermayer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Josef Teichmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Josef Teichmann.

Additional information

M. Keller-Ressel and J. Teichmann gratefully acknowledge the support from the Austrian Science Fund (FWF) under grant Y328 (START prize).

W. Schachermayer and J. Teichmann gratefully acknowledge the support from the Austrian Science Fund (FWF) under grant P19456, from the Vienna Science and Technology Fund (WWTF) under grant MA13 and by the Christian Doppler Research Association (CDG).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Keller-Ressel, M., Schachermayer, W. & Teichmann, J. Affine processes are regular. Probab. Theory Relat. Fields 151, 591–611 (2011). https://doi.org/10.1007/s00440-010-0309-4

Download citation

  • Received: 18 June 2009

  • Revised: 22 March 2010

  • Published: 30 June 2010

  • Issue Date: December 2011

  • DOI: https://doi.org/10.1007/s00440-010-0309-4

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

  • Affine processes
  • Regularity
  • Characteristic function
  • Semiflow

Mathematics Subject Classification (2000)

  • 60J25
  • 39B32
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