Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-09T00:58:43.690Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Transition Law of Tempered Stable Ornstein–Uhlenbeck Processes

Part of: Markov processes Distribution theory

Published online by Cambridge University Press:  14 July 2016

Shibin Zhang  and
Xinsheng Zhang
Shibin Zhang*
Affiliation:
Shanghai Maritime University
Xinsheng Zhang*
Affiliation:
Fudan University
*
∗Postal address: Department of Mathematics, Shanghai Maritime University, 1550 Pudong Avenue, Shanghai 200135, P. R. China. Email address: [email protected]
∗∗Postal address: Department of Statistics, Fudan University, 220 Handan Road, Shanghai 200433, P. R. China. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of two independent random variables: one has a tempered stable distribution and the other has a compound Poisson distribution. In distribution, the compound Poisson random variable is equal to the sum of a Poisson-distributed number of positive random variables, which are independent and identically distributed and have a common specified density function. Based on the representation of the stochastic integral, we prove that the transition distribution of the tempered stable Ornstein–Uhlenbeck process is self-decomposable and that the transition density is a C∞-function.

Keywords

Lévy processtempered stableOrnstein–Uhlenbeck-type processself-decomposability

MSC classification

Secondary: 60J35: Transition functions, generators and resolvents 62E15: Exact distribution theory
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 3 , September 2009 , pp. 721 - 731
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Barndorff-Nielsen, O. E. and Shephard, N. (2001). Modelling by Lévy processes for financial econometrics. In Lévy Processes, eds Barndorff-Nielsen, O. E., Mikosch, T. and Resnick, S., Birkhäuser, Boston, MA, pp. 283–318.Google Scholar
[2] Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics (with discussion). J. R. Statist. Soc. B 63, 167–241.Google Scholar
[3] Barndorff-Nielsen, O. E. and Shephard, N. (2001). Normal modified stable processes. Theory Prob. Math. Statist. 65, 1–20.Google Scholar
[4] Cariboni, C. and Schoutens, W. (2009). Jumps in intensity models: investigating the performance of Ornstein–Uhlenbeck processes in credit risk modeling. Metrika 69, 173–198.Google Scholar
[5] Chambers, J. M., Mallows, C. L. and Stuck, B.W. (1976). A method for simulating stable random variables. J. Amer. Statist. Assoc. 71, 340–344.Google Scholar
[6] Devroye, L. (1986). Nonuniform Random Variate Generation. Springer, New York.Google Scholar
[7] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
[8] Hougaard, P. (1986). Survival models for heterogeneous populations derived from stable distributions. Biometrika 73, 387–396.Google Scholar
[9] Sato, K. I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
[10] Taufer, E. and Leonenko, N. (2009). Simulation of Lévy-driven Ornstein–Uhlenbeck processes with given marginal distribution. Comput. Statist. Data Anal. 53, 2427–2437.Google Scholar
[11] Tweedie, M. C. K. (1984). An index which distinguishes between some important exponential families. In Statistics: Applications and New Directions, eds Ghosh, J. and Roy, J., Proceedings of the Indian Statistical Institute Golden Jubilee International Conference, pp. 579–604.Google Scholar
[12] Weron, R. (1996). On the Chambers-Mallows-Stuck method for simulating skewed stable random variables. Statist. Prob. Lett. 28, 165–171.Google Scholar
[13] Zhang, S. (2008). Simulation of non-Gaussian OU-based stochastic volatility models. In Proc. Internat. Symp. Financial Eng. Risk Manag., eds Ai, C. and Wu, D., Universe Academic Press, Toronto, pp. 234–238.Google Scholar
[14] Zhang, S. and Zhang, X. (2008). Exact simulation of IG-OU processes. Methodology Comput. Appl. Prob. 10, 337–355.Google Scholar
[15] Zhang, S., Zhang, X. and Sun, S. (2006). Parametric estimation of discretly sampled Gamma-OU processes. Science China Ser. A 49, 1231–1257.Google Scholar