Criteria for the Finiteness of the Strong $p$-Variation for L\'evy-type Processes

TL;DR

This paper establishes criteria for when Le9vy-type processes have finite or infinite p-variation using generalized indices, broadening previous results to more general processes and state spaces.

Contribution

It introduces a wider class of Le9vy-type processes, uses fine continuity for general state spaces, and develops a local index to determine p-variation finiteness.

Findings

Criteria for finiteness of p-variation using generalized indices

Introduction of a local index for infiniteness conditions

Applicability demonstrated through various examples

Abstract

Using generalized Blumenthal--Getoor indices, we obtain criteria for the finiteness of the $p$-variation of L\'evy-type processes. This class of stochastic processes includes solutions of Skorokhod-type stochastic differential equations (SDEs), certain Feller processes and solutions of L\'evy driven SDEs. The class of processes is wider than in earlier contributions and using fine continuity we are able to handle general measurable subsets of $R^d$ as state spaces. Furthermore, in contrast to previous contributions on the subject, we introduce a local index in order to complement the upper index. This local index yields a sufficient condition for the infiniteness of the $p$-variation. We discuss various examples in order to demonstrate the applicability of the method.