Moving average Multifractional Processes with Random Exponent: Lower bounds for local oscillations

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 Holder continuous stochastic process indexed by the integration variable. Thus, this MPRE can be expressed as a moving average Ito 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 Holder 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. (c) 2022 Elsevier B.V. All rights reserved.