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Estimating functions for SDE driven by stable Lévy processes. (English. French summary) Zbl 1467.60033

Summary: This paper is concerned with parametric inference for a stochastic differential equation driven by a pure-jump Lévy process, based on high frequency observations on a fixed time period. Assuming that the Lévy measure of the driving process behaves like that of an \(\alpha \)-stable process around zero, we propose an estimating functions based method which leads to asymptotically efficient estimators for any value of \(\alpha\in(0,2)\) and does not require any integrability assumptions on the process. The main limit theorems are derived thanks to a control in total variation distance between the law of the normalized process, in small time, and the \(\alpha \)-stable distribution. This method is an alternative to the non Gaussian quasi-likelihood estimation method proposed by H. Masuda [Stochastic Processes Appl. 129, No. 3, 1013–1059 (2019; Zbl 1450.62106)] where the Blumenthal-Getoor index \(\alpha\) is restricted to belong to the interval \([1,2)\).

MSC:

60G51 Processes with independent increments; Lévy processes
60G52 Stable stochastic processes
62F12 Asymptotic properties of parametric estimators
60H07 Stochastic calculus of variations and the Malliavin calculus
60F05 Central limit and other weak theorems

Citations:

Zbl 1450.62106
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References:

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