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.
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