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Random integral representation of operator-semi-self-similar processes with independent increments

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  • Becker-Kern, Peter

Abstract

Jeanblanc et al. (Stochastic Process. Appl. 100 (2002) 223) give a representation of self-similar processes with independent increments by stochastic integrals with respect to background driving Lévy processes. Via Lamperti's transformation these processes correspond to stationary Ornstein-Uhlenbeck processes. In the present paper we generalize the integral representation to multivariate processes with independent increments having the weaker scaling property of operator-semi-self-similarity. It turns out that the corresponding background driving process has periodically stationary increments and in general is no longer a Lévy process. Just as well it turns out that the Lamperti transform of an operator-semi-self-similar process with independent increments defines a periodically stationary process of Ornstein-Uhlenbeck type.

Suggested Citation

  • Becker-Kern, Peter, 2004. "Random integral representation of operator-semi-self-similar processes with independent increments," Stochastic Processes and their Applications, Elsevier, vol. 109(2), pages 327-344, February.
    • Handle: RePEc:eee:spapps:v:109:y:2004:i:2:p:327-344
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    References listed on IDEAS

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    1. Maejima, Makoto & Sato, Ken-iti & Watanabe, Toshiro, 2000. "Distributions of selfsimilar and semi-selfsimilar processes with independent increments," Statistics & Probability Letters, Elsevier, vol. 47(4), pages 395-401, May.
    2. Jeanblanc, M. & Pitman, J. & Yor, M., 0. "Self-similar processes with independent increments associated with Lévy and Bessel processes," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 223-231, July.
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    Cited by:

    1. Becker-Kern, Peter & Pap, Gyula, 2008. "Parameter estimation of selfsimilarity exponents," Journal of Multivariate Analysis, Elsevier, vol. 99(1), pages 117-140, January.
    2. Makoto Maejima & Taisuke Takamune & Yohei Ueda, 2014. "The Dichotomy of Recurrence and Transience of Semi-Lévy Processes," Journal of Theoretical Probability, Springer, vol. 27(3), pages 982-996, September.
    3. Saigo, Tatsuhiko & Tamura, Yozo, 2006. "Operator semi-self-similar processes and their space-scaling matrices," Statistics & Probability Letters, Elsevier, vol. 76(7), pages 675-681, April.
    4. Bhatti, T. & Kern, P., 2017. "An integral representation of dilatively stable processes with independent increments," Stochastic Processes and their Applications, Elsevier, vol. 127(1), pages 209-227.

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