# Bifractional Brownian motion for $H>1$ and $2HK\le 1$

## TL;DR

This paper proves that bifractional Brownian motion with parameters H>1 and 2HK≤1 is nonnegative definite, extending known parameter ranges, and provides a decomposition of related Gaussian processes.

## Contribution

The paper establishes nonnegative definiteness of bifractional Brownian motion for H>1 and 2HK≤1, and offers a novel decomposition of fractional Brownian motion for H<1/2.

## Key findings

- Proves nonnegative definiteness for H>1 and 2HK≤1.
- Decomposes bifractional Brownian motion into sum of simpler processes.
- Provides a decomposition of fractional Brownian motion with H<1/2.

## Abstract

Bifractional Brownian motion on $\mathbb{R}_+$ is a two parameter centered Gaussian process with covariance function: \[ R_{H,K} (t,s)=\frac 1{2^K}\left(\left(t^{2H}+s^{2H}\right)^K-\ |{t-s}\ |^{2HK}\right), \qquad s,t\ge 0. \] This process has been originally introduced by Houdr\'e and Villa (2003) for the range of parameters $H\in (0,1]$ and $K\in (0,1]$. Since then, the range of parameters, for which $R_{H,K}$ is known to be nonnegative definite has been somewhat extended, but the full range is still not known. We give an elementary proof that $R_{H,K}$ is nonnegative definite for parameters $H,K$ satisfying $H>1$ and $0<2HK\le 1$. We show that $R_{H,K}$ can be decomposed into a sum of two nonnegative definite functions. As a side product we obtain a decomposition of the fractional Brownian motion with Hurst parameter $H<\frac 12$ into a sum of time rescaled Brownian motion and another independent self-similar Gaussian process. We also discuss some simple properties of bifractional Brownian motion with $H>1$.