Some Properties of Bifractional Bessel Processes Driven by Bifractional Brownian Motion

Abstract

Let $\left(B={\left\{\left(B_t^1,\dots ,B_t^d\right)\right\}}_{t\ge 0}\right)$ be a $d$ -dimensional bifractional Brownian motion and $R_t=\sqrt{{\left(B_t^1\right)}^2+\cdots +{\left(B_t^d\right)}^2}$ be the bifractional Bessel process with the index $\left(2\text{HK}\ge 1\right)$ . The Itô formula for the bifractional Brownian motion leads to the equation $\left(R_t={\displaystyle {\sum }_{i=1}^d{\displaystyle {\int }_0^t\left(B_s^i/R_s\right)\text{d}{B}_s^i}}+\text{HK}\left(d-1\right){\displaystyle {\int }_0^t\left(s^{2\text{HK}-1}/R_s\right)\text{d}s}\right)$ . In the Brownian motion case $\left(K=1\right)$ and $\left(H=\left(1/2\right)\right)$ , $\left(X_t≔{\displaystyle {\sum }_{i=1}^d{\displaystyle {\int }_0^t\left(B_s^i/R_s\right)\text{d}{B}_s^i}},\hspace{1em}d\ge 1\right)$ is a Brownian motion by Lévy’s characterization theorem. In this paper, we prove that process $X_t$ is not a bifractional Brownian motion unless $\left(K=1\right)$ and $\left(H=\left(1/2\right)\right)$ . We also study some other properties and their application of this stochastic process.