Page 86 - Special Topic Session (STS) - Volume 3

STS517 Andreas Basse-O’Connor
more complex phenomena, which are not captured by ordinarily Ornstein–
Uhlenbeck processes. That is, we consider a process given by

where ∈ ℕ, for = 1, . . . , are independent α-stable Ornstein–

Uhlenbeck processes with parameters > 0, and ≥ 0 for = 1, . . . , are

positive numbers satisfying + ··· + = 1. Generalizing this idea we arrive

1
at the definition of superposition of Ornstein–Uhlenbeck processes (cf.
Barndorff-Nielsen (2000)):
Definition 1.1. Let denote a probability measure on ℝ+ and α ∈ (0,2]. Then
is called an α-stable superposition of Ornstein–Uhlenbeck (supOU) process
if it has a representation of the form

where Λ is a symmetric α-stable random measure on ℝ with control measure
2
().
We note that the supOU process , given in (1.4), is well-defined if and
1
only if ∫ () < ∞, cf. Fasen and Klüppelberg (2007, page 343). The
−1
0
ordinary Ornstein– Uhlenbeck process (1.2) corresponds to m = δλ, and the
more general case (1.3) corresponds to = ∑ . Throughout the paper

=1
denotes the Dirac measure at ∈ ℝ given by () = (), and ℝ : =

+

(0, ∞). An α-stable supOU process is a stationary α-stable process, however, it
is neither Markovian nor semi martingale in general, opposite to ordinary OU
processes. Moreover, supOU processes provide a flexible framework for
modelling long-range dependence, let e.g. be the (, 1)-law where ∈
(0,1), cf. Barndorff-Nielsen (2000, Example 3.1), which is also opposite to
ordinary OU processes. In the next section we will prove the main result of the
paper (Theorem 3.1), which says that if has finite − ℎ moment for some
> 0 and ∈ (1,2), then the α-stable supOU process has càdlàg sample
paths.

2. Methodology
To show that an α-stable supOU process X, given in Definition 1.1, has
càdlàg sample paths we decompose it as

() = () + (), (2.1)
where

Our treatment of the two processes and in (2.1) requires different
techniques. In particularly, under a moment condition on m, we will show, in

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