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On the Transition Law of Tempered Stable Ornstein–Uhlenbeck Processes

Part of: Distribution theory Markov processes

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: sbzhang@shmtu.edu.cn
∗∗Postal address: Department of Statistics, Fudan University, 220 Handan Road, Shanghai 200433, P. R. China. Email address: xszhang@fudan.edu.cn
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Abstract

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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 

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