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A stochastic calculus for Rosenblatt processes

Author

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  • Čoupek, Petr
  • Duncan, Tyrone E.
  • Pasik-Duncan, Bozenna

Abstract

A stochastic calculus is given for processes described by stochastic integrals with respect to fractional Brownian motions and Rosenblatt processes somewhat analogous to the stochastic calculus for Itô processes. These processes for this stochastic calculus arise naturally from a stochastic chain rule for functionals of Rosenblatt processes; and some Itô-type expressions are given here. Furthermore, there is some analysis of these results for their applications to problems using Rosenblatt noise.

Suggested Citation

  • Čoupek, Petr & Duncan, Tyrone E. & Pasik-Duncan, Bozenna, 2022. "A stochastic calculus for Rosenblatt processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 853-885.
    • Handle: RePEc:eee:spapps:v:150:y:2022:i:c:p:853-885
    DOI: 10.1016/j.spa.2020.01.004
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    References listed on IDEAS

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    1. Bardet, Jean-Marc & Tudor, Ciprian, 2014. "Asymptotic behavior of the Whittle estimator for the increments of a Rosenblatt process," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 1-16.
    2. Čoupek, P. & Maslowski, B., 2017. "Stochastic evolution equations with Volterra noise," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 877-900.
    3. Bardet, J.-M. & Tudor, C.A., 2010. "A wavelet analysis of the Rosenblatt process: Chaos expansion and estimation of the self-similarity parameter," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2331-2362, December.
    4. Nualart,David & Nualart,Eulalia, 2018. "Introduction to Malliavin Calculus," Cambridge Books, Cambridge University Press, number 9781107039124, November.
    5. Nualart,David & Nualart,Eulalia, 2018. "Introduction to Malliavin Calculus," Cambridge Books, Cambridge University Press, number 9781107611986, November.
    6. Russo, Francesco & Vallois, Pierre, 1995. "The generalized covariation process and Ito formula," Stochastic Processes and their Applications, Elsevier, vol. 59(1), pages 81-104, September.
    7. Albin, J. M. P., 1998. "A note on Rosenblatt distributions," Statistics & Probability Letters, Elsevier, vol. 40(1), pages 83-91, September.
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