  We study how to construct a stochastic process on a finite interval with
given `roughness' and finite joint moments of marginal distributions. We first
extend Ciesielski's isomorphism along a general sequence of partitions, and
provide a characterization of H\"older regularity of a function in terms of its
Schauder coefficients. Using this characterization we provide a better
(pathwise) estimator of H\"older exponent. As an additional application, we
construct fake (fractional) Brownian motions with some path properties and
finite moments of marginal distributions same as (fractional) Brownian motions.
These belong to non-Gaussian families of stochastic processes which are
statistically difficult to distinguish from real (fractional) Brownian motions.

Probability;Applications