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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aczél J.: Functional Equations and their Applications. Academic Press, New York (1966)
Dawson D.A., Li Z.: Skew convolution semigroups and affine Markov processes. Ann. Probab. 34(3), 1103–1142 (2006)
Duffie D., Filipovic D., Schachermayer W.: Affine processes and applications in finance. Ann. Appl. Probab. 13(3), 984–1053 (2003)
Filipović D., Teichmann J.: Regularity of finite-dimensional realizations for evolution equations. J. Funct. Anal. 197, 433–446 (2003)
Filipović, D., Tappe, S., Teichmann, J.: Jump-diffusions in Hilbert spaces: existence, stability and numerics, Stochastics (2010, forthcoming). arXiv/0810.5023
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
Jacob N.: Pseudo Differential Operators and Markov Processes, vol. I. Imperial College Press, London (2001)
Kawazu K., Watanabe S.: Branching processes with immigration and related limit theorems. Theory Probab. Appl. XVI(1), 36–54 (1971)
Keller-Ressel, M.: Moment explosions and long-term behavior of affine stochastic volatility models. Math. Finance (2008, forthcoming). arXiv:0802.1823
Keller-Ressel, M.: Affine processes—contributions to theory and applications. PhD thesis, TU Wien (2008)
Lukacs E.: Characteristic Functions. Charles Griffin & Co Ltd., London (1960)
Montgomery D., Zippin L.: Topological Transformation Groups. Interscience, New York (1955)
Rogers L.C.G., Williams D.: Diffusions, Markov Processes and Martingales, 2nd edn, vol. 1. Cambridge Mathematical Library, Cambridge (1994)
Semadeni Z.: Banach Spaces of Continuous Functions. Polish Scientific Publishers, Poland (1971)
Author information
Authors and Affiliations
Corresponding author
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
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-010-0309-4