  A recently proposed alternative to multifractional Brownian motion (mBm) with
random Hurst exponent is studied, which we refer to as It\^o-mBm. It is shown
that It\^o-mBm is locally self-similar. In contrast to mBm, its pathwise
regularity is almost unaffected by the roughness of the functional Hurst
parameter. The pathwise properties are established via a new polynomial moment
condition similar to the Kolmogorov-Chentsov theorem, allowing for random local
H\"older exponents. Our results are applicable to a broad class of moving
average processes where pathwise regularity and long memory properties may be
decoupled, e.g. to a multifractional generalization of the Mat\'ern process.
