Bifractional Brownian motion: existence and border cases

TL;DR

This paper investigates the conditions under which bifractional Brownian motion exists for different parameter pairs and explores related limiting processes, contributing to the understanding of this Gaussian process's properties.

Contribution

It establishes existence criteria for bifractional Brownian motion across parameter ranges and examines associated limiting processes, advancing theoretical knowledge.

Findings

Existence conditions for bfBm depending on (H,K) parameters

Identification of related limiting processes in border cases

Extension of understanding of Gaussian processes with complex covariance structures

Abstract

Bifractional Brownian motion (bfBm) is a centered Gaussian process with covariance \[ R^{(H,K)}(s,t)= 2^{-K} \left( \left(|s|^{2H}+|t|^{2H} \right)^{K}-|t-s|^{2HK}\right), \qquad s,t\in R. \] We study the existence of bfBm for a given pair of parameters $(H,K)$ and encounter some related limiting processes.