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Multifractional processes are extensions of Fractional Brownian Motion obtained by replacing its constant Hurst parameter by a deterministic or a random function H(·), called the Hurst function, which allows one to prescribe the local roughness of sample paths at each point. For that reason statistical estimation of H(·) is an important issue. Many articles have dealt with this issue in the case where H(·) is deterministic. However, statistical estimation of H(·) when it is random remains an open problem. The main goal of our present article is to propose, under a weak local Hölder regularity condition on H(·), a solution for this problem in the framework of Moving Average Multifractional Process with Random Exponent (MAMPRE), denoted by X. From the data consisting in a discrete realization of X on the interval [0; 1], we construct a continuous piecewise linear random function which almost surely converges to H(·) for the uniform norm, when the size of the discretization mesh goes to zero; we also provide an almost sure estimate of the uniform rate of convergence and we explain how it can be optimized. Such kind of strong consistency result in uniform norm is rather unusual in literature on statistical estimation of functions. [ABSTRACT FROM AUTHOR]

In the last few years Ayache, Esser and Hamonier introduced a new Multifractional Process with Random Exponent (MPRE) obtained by replacing the Hurst parameter in a moving average representation of Fractional Brownian Motion through Wiener integral by an adapted Hölder continuous stochastic process indexed by the integration variable. Thus, this MPRE can be expressed as a moving average Itô integral which is a considerable advantage with respect to another MPRE introduced a long time ago by Ayache and Taqqu. Thanks to this advantage, very recently, Loboda, Mies and Steland have derived interesting results on local Hölder regularity, self-similarity and other properties of the recently introduced moving average MPRE and generalizations of it. Yet, the problem of obtaining, on a universal event of probability 1 not depending on the location, relevant lower bounds for local oscillations of such processes has remained open. We solve it in the present article under some conditions. [ABSTRACT FROM AUTHOR]

A first type of Multifractional Process with Random Exponent (MPRE) was constructed several years ago in [Publ. Mat.49 (2005), 459–486] by replacing in a wavelet series representation of Fractional Brownian Motion (FBM) the Hurst parameter by a random variable depending on the time variable. In the present article, we propose another approach for constructing another type of MPRE. It consists in substituting to the Hurst parameter, in a stochastic integral representation of the high-frequency part of FBM, a random variable depending on the integration variable. The MPRE obtained in this way offers, among other things, the advantages to have a representation through classical Itô integral and to be less difficult to simulate than the first type of MPRE, previously introduced in [Publ. Mat.49 (2005), 459–486]. Yet, the study of Hölder regularity of this new MPRE is a significantly more challenging problem than in the case of the previous one. Actually, it requires to develop a new methodology relying on an extensive use of the Haar basis. [ABSTRACT FROM AUTHOR]