Page 89 - Special Topic Session (STS) - Volume 3

STS517 Andreas Basse-O’Connor

Choose ∈ [0,1] such that 1/ < ≤ ( + 1)/, which is possible since
> 0 and > 1.
By the inequality

and (2.12) we deduce that

where we have used that {||,1} ≤ || for all ∈ ℝ, in the third inequality
(recall ∈ [0,1]), and that 0 < − 1 ≤ in the last inequality together
with the fact that m is a finite measure. Hence,

‖() − ()‖ ≤ | − | . (2.13)

Fix > 0 and consider the metric (, ) = ‖() − ()‖ on [0, ]

induced by the −stable process . For each > 0 we let () denote the
smallest number of open −balls of radius needed to cover [0, ]. From the
estimate (2.13) we deduce that

and hence

where the last inequality follows by the fact that > 1/. By (2.14) and
Theorem 12.2.1 of Samorodnitsky and Taqqu (1994) we deduce that has
continuous sample paths with probability one, which concludes the proof.

3. Result
In this section we state the main result of the paper, which gives sufficient
conditions for stable supOU processes to have càdlàg sample paths.
Theorem 3.1. Let be an −stable supOU process given in Definition 1.1
with ∈ (1,2) and such that the measure m has finite − ℎ moment for
some > 0. Then has c`adl`ag sample paths with probability one.
Proof. The theorem follows by the decomposition (2.1) and Lemmas 2.1 and
2.2.

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