A unifying approach to fractional Lévy processes

Abstract

Starting from the moving average representation of fractional Brownian motion fractional Lévy processes have been constructed by keeping the same moving average kernel and replacing the Brownian motion by a pure jump Lévy process with finite second moments. Another way was to replace the Brownian motion by an alpha-stable Lévy process and the exponent in the kernel by H-1/alpha. We now provide a unifying approach taking kernels of the form a((t-s)_+^gamma - (-s)_+^gamma) + b((t-s)_-^gamma - (-s)_-^gamma), where gamma can be chosen according to the existing moments and the Blumenthal-Getoor index of the underlying Lévy process. These processes may exhibit both long and short range dependence. In addition we will examine further properties of the processes, e.g. regularity of the sample paths and the semimartingale property. MSC 2010: 60G22, 60E07