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

Abstract

We study a real-valued L\'evy-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 L\'evy kernel and unboundedness of the drift coefficient.