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2019 Approximation in law of locally $\alpha $-stable Lévy-type processes by non-linear regressions
Alexei Kulik
Electron. J. Probab. 24: 1-45 (2019). DOI: 10.1214/19-EJP339


We study a real-valued Lévy-type process $X$, which is locally $\alpha $-stable in the sense that its jump kernel is a combination of a ‘principal’ (state dependent) $\alpha $-stable part with a ‘residual’ lower order part. We show that under mild conditions on the local characteristics of a process (the jump kernel and the velocity field) the process is uniquely defined, is Markov, and has the strong Feller property. We approximate $X$ in law by a non-linear regression $\widetilde{X} ^{x}_{t}=\mathfrak{f} _{t}(x)+t^{1/\alpha }U^{x}_{t} $ with a deterministic regressor term $\mathfrak{f} _{t}(x)$ and $\alpha $-stable innovation term $U^{x}_{t}$, and provide error estimates for such an approximation. A case study is performed, revealing different types of assumptions which lead to various choices of regressor/innovation terms and various types of the estimates. The assumptions are quite general, cover the super-critical case $\alpha <1$, and allow non-symmetry of the Lévy kernel and unboundedness of the drift coefficient.


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Alexei Kulik. "Approximation in law of locally $\alpha $-stable Lévy-type processes by non-linear regressions." Electron. J. Probab. 24 1 - 45, 2019.


Received: 23 January 2019; Accepted: 7 July 2019; Published: 2019
First available in Project Euclid: 10 September 2019

zbMATH: 07107390
MathSciNet: MR4003136
Digital Object Identifier: 10.1214/19-EJP339

Primary: 35S05, 35S10, 47G30, 60J35, 60J75


Vol.24 • 2019
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