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Operator-stable-like Processes

Peter Scheffler, Alexander Schnurr, Daniel Schulte

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.

Operator-stable-like Processes

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 -variation. The class introduced here includes stable-like processes as special case.

Paper Structure

This paper contains 8 sections, 37 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Construction of Operator-stable-like Processes
  3. Properties of Operator-stable-like Processes
  4. Maximal inequalities
  5. Moments
  6. Asymptotic Properties
  7. $p$-Variation
  8. Appendix: Matrix exponential, generalized polar coordinates and estimates

Key Result

Lemma 2.6

For all measurable $f: \Gamma \rightarrow \mathbb{R}_+$ it holds that:

Theorems & Definitions (77)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • Proposition 2.7
  • proof
  • Example 2.8
  • Theorem 2.9
  • ...and 67 more