An extension of the standard multifractional Brownian motion

In this paper, firstly, we generalize the definition of the bifractional Brownian motion $B^{H,K}:=(B^{H,K};t\ge 0)$, with parameters $H\in (0,1)$ and $K\in (0,1]$, to the case where $H$ is no longer a constant, but a function $H(\mathrm{.})$ of the time index $t$ of the process. We denote this new process by $B^{H(\mathrm{.}),K}$. Secondly, we study its time regularities, the local asymptotic self-similarity and the long-range dependence properties. {\bf Key words:} {Gaussian process; Self similar process; Fractional Brownian motion; Bifractional Brownian motion; Multifractional Brownian motion; Local asymptotic self-similarity.}