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An integral representation of dilatively stable processes with independent increments

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  • Bhatti, T.
  • Kern, P.

Abstract

Dilative stability generalizes the property of selfsimilarity for infinitely divisible stochastic processes by introducing an additional scaling in the convolution exponent. Inspired by results of Iglói (2008), we will show how dilatively stable processes with independent increments can be represented by integrals with respect to time-changed Lévy processes. Via a Lamperti-type transformation these representations are shown to be closely connected to translatively stable processes of Ornstein–Uhlenbeck-type, where translative stability generalizes the notion of stationarity. The presented results complement corresponding representations for selfsimilar processes with independent increments known from the literature.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:1:p:209-227
    DOI: 10.1016/j.spa.2016.06.006
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    References listed on IDEAS

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    1. Kern, Peter & Wedrich, Lina, 2015. "Dilatively semistable stochastic processes," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 101-108.
    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.
    3. Es-sebaiy, Khalifa & Ouknine, Youssef, 2008. "How rich is the class of processes which are infinitely divisible with respect to time?," Statistics & Probability Letters, Elsevier, vol. 78(5), pages 537-547, April.
    4. Wolfe, Stephen James, 1982. "On a continuous analogue of the stochastic difference equation Xn=[rho]Xn-1+Bn," Stochastic Processes and their Applications, Elsevier, vol. 12(3), pages 301-312, May.
    5. 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.
    6. Pilipauskaitė, Vytautė & Surgailis, Donatas, 2014. "Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes," Stochastic Processes and their Applications, Elsevier, vol. 124(2), pages 1011-1035.
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