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Stochastic evolution equations with Volterra noise

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  • Čoupek, P.
  • Maslowski, B.

Abstract

Volterra processes are continuous stochastic processes whose covariance function can be written in the form R(s,t)=∫0s∧tK(s,r)K(t,r)dr, where K is a suitable square integrable kernel. Examples of such processes are the fractional Brownian motion, multifractional Brownian motion or (in the non-Gaussian case) Rosenblatt process. In the first part, stochastic integral with respect to Volterra processes and cylindrical Volterra process in Hilbert spaces are defined and some of their properties are studied. In the second part, these results are applied to linear stochastic equations in Hilbert spaces driven by cylindrical Volterra processes. Measurability, mean-square continuity and paths continuity of their solutions are proved under various sets of conditions. The general results are illustrated by examples of parabolic and hyperbolic SPDEs.

Suggested Citation

  • Čoupek, P. & Maslowski, B., 2017. "Stochastic evolution equations with Volterra noise," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 877-900.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:3:p:877-900
    DOI: 10.1016/j.spa.2016.07.003
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    References listed on IDEAS

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    1. Hult, Henrik, 2003. "Approximating some Volterra type stochastic integrals with applications to parameter estimation," Stochastic Processes and their Applications, Elsevier, vol. 105(1), pages 1-32, May.
    2. Mémin, Jean & Mishura, Yulia & Valkeila, Esko, 2001. "Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 51(2), pages 197-206, January.
    3. Grecksch, W. & Anh, V. V., 1999. "A parabolic stochastic differential equation with fractional Brownian motion input," Statistics & Probability Letters, Elsevier, vol. 41(4), pages 337-346, February.
    4. Lebovits, Joachim & Lévy Véhel, Jacques & Herbin, Erick, 2014. "Stochastic integration with respect to multifractional Brownian motion via tangent fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 678-708.
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    Cited by:

    1. Dhayal, Rajesh & Malik, Muslim, 2021. "Approximate controllability of fractional stochastic differential equations driven by Rosenblatt process with non-instantaneous impulses," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    2. 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.
    3. Obayda Assaad & Ciprian A. Tudor, 2020. "Parameter identification for the Hermite Ornstein–Uhlenbeck process," Statistical Inference for Stochastic Processes, Springer, vol. 23(2), pages 251-270, July.

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