Page 90 - Special Topic Session (STS) - Volume 3

STS517 Andreas Basse-O’Connor
4. Discussion and Conclusion
In the following we fix ∈ (1,2) and let be an −stable supOU process
given in Definition 1.1. We recall that due to the discontinuity of the integrand
↦ −(−) (−∞,] () in (1.4) at = , the process will never have
continuous sample paths, cf. Rosiński (1989). However, the above Theorem 3.1
shows that if has finite − ℎ moment for some > 0 then has càdlàg
sample paths with probability one. To compare this condition, with the
literature we note that Example 4.1 in Basse-O’Connor and Rosiński (2016),
shows that is a semi martingale if and only if has finite ( − 1) − ℎ
moment, and if is a semi martingale then it has càdlàg sample paths. Since
− 1 > 0 in our setting, the conditions in Theorem 3.1 are weaker than the
ones we can derive from Basse-O’Connor and Rosiński (2016).

References
1. Barndorff-Nielsen, O. E. (2000). Superposition of Ornstein-Uhlenbeck type
processes. Teor. Veroyatnost. i Primenen. 45(2), 289–311.
2. Basse-O’Connor, A. and J. Rosiński (2013). On the uniform convergence of
random series in Skorohod space and representations of c`adl`ag infinitely
divisible processes. Ann. Probab. 41(6), 4317–4341.
3. Basse-O’Connor, A. and J. Rosiński (2016). On infinitely divisible
semimartingales. Probab. Theory Related Fields 164(1-2), 133–163.
4. Fasen, V. and C. Klüppelberg (2007). Extremes of supOU processes. In
Stochastic analysis and applications, Volume 2 of Abel Symp., pp. 339–
359. Springer, Berlin.
5. Rosiński, J. (1989). On path properties of certain infinitely divisible
processes. Stochastic Process. Appl. 33(1), 73–87.
6. Samorodnitsky, G. and M. S. Taqqu (1994). Stable Non-Gaussian Random
Processes. Stochastic Modeling. New York: Chapman & Hall. Stochastic
models with infinite variance.

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