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

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 Levy 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 Levy 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 Levy processes.