Maximal inequalities for fractional L\'evy and related processes

TL;DR

This paper establishes maximal inequalities for convoluted martingales driven by centered Lévy processes, including fractional Lévy processes, showing their maximum processes are p-integrable under certain conditions.

Contribution

It provides new maximal inequalities for fractional Lévy and related processes, extending existing results to a broader class of convoluted martingales.

Findings

Maximum process of convoluted martingale is p-integrable

Derived maximal inequalities depend on kernel and martingale moments

Applicable to fractional Lévy processes and similar models

Abstract

In this paper we study processes which are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred L\'evy process, which covers the popular class of fractional L\'evy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding `convoluted martingale' is $p$-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.