Approximation in law of locally $\alpha $-stable Lévy-type processes by non-linear regressions

Abstract

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.