Fractional L\'evy processes as a result of compact interval integral transformation

TL;DR

This paper introduces a new way to define fractional Le9vy processes using compact interval integrals, compares them to fractional Brownian motion, and explores their properties with applications in finance.

Contribution

It provides a novel compact interval integral representation of fractional Le9vy processes and establishes conditions for their equivalence to fractional Brownian motion.

Findings

Fractional Le9vy processes via different integral transforms share the same distributions only if they are fractional Brownian motions.

Relations between various fractional Le9vy processes are characterized.

Properties of these processes are analyzed, including a financial application.

Abstract

Fractional Brownian motion can be represented as an integral of a deterministic kernel w.r.t. an ordinary Brownian motion either on infinite or compact interval. In previous literature fractional L\'evy processes are defined by integrating the infinite interval kernel w.r.t. a general L\'evy process. In this article we define fractional L\'evy processes using the compact interval representation. We prove that the fractional L\'evy processes presented via different integral transformations have the same finite dimensional distributions if and only if they are fractional Brownian motions. Also, we present relations between different fractional L\'evy processes and analyze the properties of such processes. A financial example is introduced as well.