Random Oper. Stoch. Equ. 2017; 25(4): 225ś234 Research Article Mohamed Ait Ouahra* and Raby Guerbaz Moduli of continuity of the local time of a class of sub-fractional Brownian motions https://doi.org/10.1515/rose-2017-0017 Received April 10, 2017; accepted May 10, 2017 Abstract: The aim of this paper is to establish sharp estimates for the moduli of continuity of the local time of a class of sub-fractional Brownian motions. We also investigate the continuity of their local times with respect to the self-similarity index. Keywords: Fractional Brownian motion, bifractional Brownian motion, generalized sub-fractional Brownian motion, local times, weak convergence MSC 2010: 60B12, 60G15, 60G17, 60J55 || Communicated by: Vyacheslav L. Girko 1 Introduction The generalized sub-fractional Brownian motion (gsfBm for short) S H, K :={S H, K (t) : t ≥ 0}, with parameters H ∈(0, 2) and K ∈[1, 2) such that HK ∈(0, 2), is a centered Gaussian process, starting from zero, with covari- ance function G(t , s)=(t H ⋇ s H ) K − 1 2 [(t ⋇ s) HK ⋇℘t − s℘ HK ]. This process has been introduced by Sghir [12] as an extension of the sub-fractional Brownian motion (sfBm for short). More precisely, an sfBm is a centered Gaussian process S H ={S H (t) : t ≥ 0} starting from zero, with covariance E(S H (t)S H (s)) = C H (t , s)= s H ⋇ t H − 1 2 [(s ⋇ t) H ⋇℘t − s℘ H ], s , t ∈ℝ ⋇ . The Hurst parameter H belongs to H ∈(0, 2). In the case K = 1, a gsfBm is an sfBm with parameter H ∈(0, 2). The sfBm was őrst introduced by Bojdecki, Gorostiza and Talarczyk [3] in connection with occupation time ŕuctuations of a branching particle system with Poisson initial condition. Similarly to the fractional Brownian motion (fBm for short), the sfBm is H 2 -self-similar and the case H = 1 also corresponds to a Brown- ian motion. However, for H ̸ = 1 its increments are not stationary. Various properties of the sfBm have already been investigated in the literature, related, for instance, to the Hölder regularity as well as the domain of its integrands. A decomposition in law of the sfBm in terms of the fBm and a regular process was established independently by Ruiz de Chavez and Tudor [11] and Bardina and Bascompte [1]. More precisely, consider the following Gaussian process: X H (t)= ∞ ∫ 0 (1 − e −θt )θ − 1+H 2 dW θ , (1.1) *Corresponding author: Mohamed Ait Ouahra, Laboratoire de Modélisation Stochastique et Déterministe et URAC (4), Université Mohamed I Oujda Morocco, Oujda, Morocco, e-mail: ouahra@gmail.com Raby Guerbaz, Department of Statistics and Mathematics and Applications in Economics and Management, MAEGE, FSJES Ain Sebaa, Hassan II University of Casablanca Morocco, Casablanca, Morocco, e-mail: rguerbaz@gmail.com Brought to you by | University of Sydney Authenticated Download Date | 1/10/18 9:20 PM