Criteria for the finiteness of the strong p-variation for Lévy-type processes

Abstract

Using generalized Blumenthal-Getoor indices, we obtain criteria for the finiteness of the p-variation of Lévy-type processes. This class of stochastic processes includes solutions of Skorokhod-type stochastic differential equations (SDEs), certain Feller processes and solutions of Lévy 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.

Noise reinforcement for Lévy processes

“…The latter was introduced in [4], and it is well-known that several local path properties of ξ are then similar to those of a β-stable Lévy process. We refer to the introduction in [16]f o r some background and historical perspective on this notion. We shall see that p c = 1/β is the critical memory parameter for noise reinforcement of Lévy processes, specificallyξ can be defined when the memory parameter p is admissible, that is smaller than 1/β, but this is no longer possible when pβ > 1.…”

The fourth characteristic of a semimartingale

“…Since the symbol contains the same information as the characteristics it has been used (in the conservative case) to analyze e.g. the Hausdorff dimension of paths [29], their strong variation [24], Hölder conditions [30], ultracontractivity of semigroups [32], laws of iterated logarithm [33] and stationary distributions of Markov processes [1].…”

A criterion for invariant measures of Itô processes based on the symbol

“…For the even wider class of Hunt semimartingales, the limit (2.1) is not defined and hence the symbol does not exist any more (cf. [19]).…”