Grey Brownian motion local time: Existence and weak-approximation

TL;DR

This paper studies grey Brownian motion, establishing the existence of local time and its weak approximation, and explores various representations and convergence properties of the process.

Contribution

It demonstrates the existence of local time for grey Brownian motion and introduces a weak-approximation method using crossings of convolution approximations.

Findings

Grey Brownian motion admits multiple representations.

The local time exists and is square integrable.

Local time can be approximated by crossing counts of approximated processes.

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.