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 Hurst parameter in moving average representation of Fractional Brownian Motion through Wiener integral an adapted Hölder continuous stochastic process indexed integration variable. Thus, this MPRE can be expressed as Itô which is considerable advantage respect to another long time ago Ayache Taqqu. Thanks advantage, very recently, Loboda, Mies Steland have derived interesting results on local regularity, self-similarity other properties recently generalizations it. Yet, problem obtaining, universal event probability 1 not depending location, relevant lower bounds for oscillations such processes has remained open. We solve it present article under some conditions.