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Lévy-driven Volterra Equations in Space and Time

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  • Carsten Chong

    (Technische Universität München)

Abstract

We investigate nonlinear stochastic Volterra equations in space and time that are driven by Lévy bases. Under a Lipschitz condition on the nonlinear term, we give existence and uniqueness criteria in weighted function spaces that depend on integrability properties of the kernel and the characteristics of the Lévy basis. Particular attention is devoted to equations with stationary solutions, or more generally, to equations with infinite memory, that is, where the time domain of integration starts at minus infinity. Here, in contrast to the case where time is positive, the usual integrability conditions on the kernel are no longer sufficient for the existence and uniqueness of solutions, but we have to impose additional size conditions on the kernel and the Lévy characteristics. Furthermore, once the existence of a solution is guaranteed, we analyze its asymptotic stability, that is, whether its moments remain bounded when time goes to infinity. Stability is proved whenever kernel and characteristics are small enough, or the nonlinearity of the equation exhibits a fractional growth of order strictly smaller than one. The results are applied to the stochastic heat equation for illustration.

Suggested Citation

  • Carsten Chong, 2017. "Lévy-driven Volterra Equations in Space and Time," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1014-1058, September.
    • Handle: RePEc:spr:jotpro:v:30:y:2017:i:3:d:10.1007_s10959-015-0662-4
    DOI: 10.1007/s10959-015-0662-4
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    References listed on IDEAS

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    1. Wang, Zhidong, 2008. "Existence and uniqueness of solutions to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients," Statistics & Probability Letters, Elsevier, vol. 78(9), pages 1062-1071, July.
    2. Reiß, M. & Riedle, M. & van Gaans, O., 2006. "Delay differential equations driven by Lévy processes: Stationarity and Feller properties," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1409-1432, October.
    3. Cochran, W. George & Lee, Jung-Soon & Potthoff, Jürgen, 1995. "Stochastic Volterra equations with singular kernels," Stochastic Processes and their Applications, Elsevier, vol. 56(2), pages 337-349, April.
    4. Albeverio, Sergio & Wu, Jiang-Lun & Zhang, Tu-Sheng, 1998. "Parabolic SPDEs driven by Poisson white noise," Stochastic Processes and their Applications, Elsevier, vol. 74(1), pages 21-36, May.
    5. Yacine Aït-Sahalia & Jean Jacod, 2014. "High-Frequency Financial Econometrics," Economics Books, Princeton University Press, edition 1, number 10261.
    6. Mueller, Carl, 1998. "The heat equation with Lévy noise," Stochastic Processes and their Applications, Elsevier, vol. 74(1), pages 67-82, May.
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    Cited by:

    1. Tomasz Kosmala & Markus Riedle, 2021. "Stochastic Integration with Respect to Cylindrical Lévy Processes by p-Summing Operators," Journal of Theoretical Probability, Springer, vol. 34(1), pages 477-497, March.

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