Grey Brownian motion local time: Existence and weak-approximation

Abstract

In this paper we investigate the class of grey Brownian motions $B_{α,β}$ ($0<α<2$, $0<β\leq1$). We show that grey Brownian motion admits different representations in terms of certain known processes, such as fractional Brownian motion, multivariate elliptical distribution or as a subordination. The weak convergence of the increments of $B_{α,β}$ in $t$, $w$-variables are studied. Using the Berman criterium we show that $B_{α,β}$ admits a $λ$-square integrable local time $L^{B_{α,β}}(\cdot,I)$ almost surely ($λ$ Lebesgue measure). Moreover, we prove that this local time can be weak-approximated by the number of crossings $C^{B_{α,β}^{\varepsilon}}(x,I)$, of level $x$, of the convolution approximation $B_{α,β}^{\varepsilon}$ of grey Brownian motion.