Grey Brownian motion local time: Existence and weak-approximation

Abstract

In this paper we investigate the class of grey Brownian motions $B_{\alpha,\beta}$ ($0<\alpha<2$, $0<\beta\leq1$). 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 $B_{\alpha,\beta}$ in $t$, $w$-variables are studied. Using the Berman criterium we show that $B_{\alpha,\beta}$ admits a $\lambda$-square integrable local time $L^{B_{\alpha,\beta}}(\cdot,I)$ almost surely ($\lambda$ Lebesgue measure). Moreover, we prove that this local time can be weak-approximated by the number of crossings $C^{B_{\alpha,\beta}^{\varepsilon}}(x,I)$, of level $x$, of the convolution approximation $B_{\alpha,\beta}^{\varepsilon}$ of grey Brownian motion.