Operator-stable-like Processes

TL;DR

This paper introduces operator-stable-like processes that locally resemble operator-stable processes but are not necessarily spatially homogeneous, analyzing their existence, path behavior, moments, and variation.

Contribution

It presents the first construction and analysis of operator-stable-like processes, extending stable-like processes to a broader class with non-homogeneous properties.

Findings

Existence of operator-stable-like processes established

Maximal estimates and moment conditions derived

Sample path behavior and p-variation analyzed

Abstract

In the present paper, we introduce so-called operator-stable-like processes. Roughly speaking, they behave locally like operator-stable processes, but they need not to be homogenous in space. Having shown existence for this class of processes, we analyze maximal estimates, the existence of moments, the short- and long-time behavior of the sample paths and $p$-variation. The class introduced here includes stable-like processes as special case.