Fractional martingales and characterization of the fractional Brownian motion

TL;DR

This paper introduces fractional martingales as derivatives of local martingales and extends Lévy's characterization to fractional Brownian motion, revealing new properties and behaviors of these stochastic processes.

Contribution

It defines fractional martingales via fractional derivatives and extends classical characterization theorems to fractional Brownian motion.

Findings

Fractional martingales have finite variation of specific order.

Extended Lévy's characterization to fractional Brownian motion.

Established properties of fractional derivatives of local martingales.

Abstract

In this paper we introduce the notion of fractional martingale as the fractional derivative of order $\alpha$ of a continuous local martingale, where $\alpha\in(-{1/2},{1/2})$, and we show that it has a nonzero finite variation of order $\frac{2}{1+2\alpha}$, under some integrability assumptions on the quadratic variation of the local martingale. As an application we establish an extension of L\'evy's characterization theorem for the fractional Brownian motion.