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

Anna Talarczyk

Published 2019-02-25Version 1

Bifractional Brownian motion on $\mathbb{R}_+$ is a two parameter centered Gaussian process with covariance function: $$ R_{H,K} (t,s)=\frac 1{2^K}\left(\left(t^{2H}+s^{2H}\right)^K-\vert t-s\vert^{2HK}\right), \qquad s,t\ge 0. $$ This process has been originally introduced by Houdr\'e and Villa (2002) 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. In the present paper 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 also 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. The main idea relies on decomposition of $R_{H,K}$ with help of the following two functions $$ C_{\gamma}(s,t)=(s+t)^\gamma-\left(\max (s,t)\right)^\gamma, \qquad s,t\ge 0 $$ and $$ Q_{\gamma}(s,t)=\left(\max (s,t)\right)^\gamma-\vert t-s\vert^\gamma, \qquad s,t\ge 0 $$ which are nonnegative definite for $0<\gamma\le 1$. We also discuss some simple properties of bifractional Brownian motion with $H>1$.