An extension of bifractional Brownian motion

TL;DR

This paper introduces a new class of self-similar Gaussian processes extending bifractional Brownian motion with parameters in different ranges, highlighting their semimartingale properties when specific conditions are met.

Contribution

It extends the bifractional Brownian motion framework to new parameter ranges, analyzing its self-similarity and semimartingale characteristics.

Findings

The process is self-similar with parameters in (0,1) for H and (1,2) for K.

It is a semimartingale when 2HK=1.

The process exhibits different properties compared to the classical bifractional Brownian motion.

Abstract

In this paper we introduce and study a self-similar Gaussian process that is the bifractional Brownian motion $B^{H,K}$ with parameters $H\in (0,1)$ and $K\in(1,2)$ such that $HK\in(0,1)$. A remarkable difference between the case $K\in(0,1)$ and our situation is that this process is a semimartingale when $2HK=1$.