Grey Brownian motion local time: Existence and weak-approximation

Abstract

In this paper we investigate the class of grey Brownian motions $B_{\alpha ,\beta }$ ($0<\alpha <2$, $0<\beta \le 1$). 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 $C^{B_{\alpha ,\beta }^{\epsilon }}(x,I)$, of level $x$, of the convolution approximation $B_{\alpha ,\beta }^{\epsilon }$ of grey Brownian motion.

20 pages

References in corpus (3)

• Characterizations and simulations of a class of stochastic processes to model anomalous diffusion
• Level crossings and other level functionals of stationary Gaussian processes
• $L^p$ moduli of continuity of Gaussian processes and local times of symmetric Lévy processes

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