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

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Abstract

Consider a random process s that is a solution of the stochastic differential equation \(\mathrm {L}s = w\) with \(\mathrm {L}\) 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 \(\mathrm {L}\) and w such that \(a^H s(\cdot / a)\) converges in law to a non-trivial self-similar process for some H, when \(a \rightarrow 0\) (coarse-scale behavior) or \(a \rightarrow \infty \) (fine-scale behavior). The parameter H depends on the homogeneity order of the operator \(\mathrm {L}\) 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.

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Acknowledgements

The authors are grateful to Thomas Humeau for fruitful discussions that led to this work, in particular concerning Proposition 3.5. We also warmly thank Virginie Uhlmann for her help with the simulations. The research leading to these results was funded by the ERC grant agreement No 692726 - FUN-SP.

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  1. École polytechnique fédérale de Lausanne, Biomedical Imaging Group, 1015, Lausanne, Switzerland

    Julien Fageot & Michael Unser

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  1. Julien Fageot
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Correspondence to Julien Fageot.

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Fageot, J., Unser, M. Scaling Limits of Solutions of Linear Stochastic Differential Equations Driven by Lévy White Noises. J Theor Probab 32, 1166–1189 (2019). https://doi.org/10.1007/s10959-018-0809-1

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