Approximation in law of locally α-stable Lévy-type processes by non-linear regressions

2018/08/21 by Kulik, Alexei
#60J35 60J75 35S05 35S10 47G30 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1808.06779

Abstract

We study a real-valued Lévy-type process X, which is locally α-stable in the sense that its jump kernel is a combination of a `principal' (state dependent) α-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=\mathfrakf_t(x)+t^1/αU^x_t with a deterministic regressor term \mathfrakf_t(x) and α-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 α<1, and allow non-symmetry of the Lévy kernel and unboundedness of the drift coefficient.

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