Path Properties of a Generalized Fractional Brownian Motion

TL;DR

This paper investigates the sample path properties of the generalized fractional Brownian motion, a Gaussian process with self-similarity and non-stationary increments, focusing on regularity, differentiability, and probabilistic laws.

Contribution

It provides a comprehensive analysis of path properties for the generalized fractional Brownian motion, extending understanding beyond classical models.

Findings

Establishes Holder continuity conditions.

Determines conditions for path differentiability.

Derives functional and local Law of the Iterated Logarithms.

Abstract

The generalized fractional Brownian motion is a Gaussian self-similar process whose increments are not necessarily stationary. It appears in applications as the scaling limit of a shot noise process with a power law shape function and non-stationary noises with a power-law variance function. In this paper we study sample path properties of the generalized fractional Brownian motion, including Holder continuity, path differentiability/non-differentiability, and functional and local Law of the Iterated Logarithms.