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Scaling Limits of Solutions of Linear Stochastic Differential Equations Driven by Lévy White Noises

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  • Julien Fageot

    (Biomedical Imaging Group)

  • Michael Unser

    (Biomedical Imaging Group)

Abstract

Consider a random process s that is a solution of the stochastic differential equation $$\mathrm {L}s = w$$ L s = w with $$\mathrm {L}$$ L a homogeneous operator and w a multidimensional Lévy white noise. In this paper, we study the asymptotic effect of zooming in or zooming out of the process s. More precisely, we give sufficient conditions on $$\mathrm {L}$$ L and w such that $$a^H s(\cdot / a)$$ a H s ( · / a ) converges in law to a non-trivial self-similar process for some H, when $$a \rightarrow 0$$ a → 0 (coarse-scale behavior) or $$a \rightarrow \infty $$ a → ∞ (fine-scale behavior). The parameter H depends on the homogeneity order of the operator $$\mathrm {L}$$ L and the Blumenthal–Getoor and Pruitt indices associated with the Lévy white noise w. Finally, we apply our general results to several famous classes of random processes and random fields and illustrate our results on simulations of Lévy processes.

Suggested Citation

  • Julien Fageot & Michael Unser, 2019. "Scaling Limits of Solutions of Linear Stochastic Differential Equations Driven by Lévy White Noises," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1166-1189, September.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:3:d:10.1007_s10959-018-0809-1
    DOI: 10.1007/s10959-018-0809-1
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    References listed on IDEAS

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    1. Fageot, Julien & Fallah, Alireza & Unser, Michael, 2017. "Multidimensional Lévy white noise in weighted Besov spaces," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1599-1621.
    2. Breton, Jean-Christophe & Dombry, Clément, 2009. "Rescaled weighted random ball models and stable self-similar random fields," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3633-3652, October.
    3. Deng, Chang-Song & Schilling, René L., 2015. "On shift Harnack inequalities for subordinate semigroups and moment estimates for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 125(10), pages 3851-3878.
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