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Gaussian asymptotics for a non-linear Langevin type equation driven by an \(\alpha\)-stable Lévy noise. (English) Zbl 1328.60135

Summary: Consider the dynamics of a particle whose speed satisfies a one-dimensional stochastic differential equation driven by a small symmetric \(\alpha\)-stable Lévy process in a potential of the form of a power function of exponent \(\beta+1\). Two cases are studied: the noise could be path continuous, namely a standard Brownian motion, if \(\alpha=2\), or pure jump Lévy process, if \(\alpha\in(0,2)\). The main goal is to study a scaling limit of the position process with this speed, and one proves that the limit is Brownian in either case. This result is a generalization in some sense of the quadratic potential case studied recently by R. Hintze and I. Pavlyukevich [Bernoulli 20, No. 1, 265–281 (2014; Zbl 1309.60059)].

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60F17 Functional limit theorems; invariance principles
60G51 Processes with independent increments; Lévy processes
60G52 Stable stochastic processes
60F05 Central limit and other weak theorems
60G44 Martingales with continuous parameter
60J65 Brownian motion
60J75 Jump processes (MSC2010)

Citations:

Zbl 1309.60059
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