Gaussian asymptotics for a non-linear Langevin type equation driven by an $\alpha$-stable Lévy noise
Résumé
Consider a one-dimensional process $x^{\varepsilon}_{t}$ the position of a particle at time $t$ which speed $v^{\varepsilon}_{t}$ is a solution of a stochastic differential equation driven by a small $\alpha$-stable Lévy process, $\varepsilon\ell_t$, $\alpha\in(0,2]$, and with a non-linear drift coefficient $-{\rm sgn}(v)|v|^{\beta}$, $\beta>2-(\frac{\alpha}{2})$. The noise could be path continuous (Brownian motion $\alpha=2$) or pure jump process ($0<\alpha<2$). We prove that, as $\varepsilon$ goes to 0, the limit in distribution of the process $\{\varepsilon^{(\beta+(\alpha/2)-2)\theta_{\alpha,\beta}}\,x^{\varepsilon}_{\varepsilon^{-\alpha}t}:t\geq 0\}$ is a Brownian motion with some variance $\kappa_{\alpha,\beta}$, where $\theta_{\alpha,\beta}=\frac{\alpha}{(\beta+\alpha-1)}$. This result is a generalization in some sense of the linear case studied by Hintze and Pavlyukevich.
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