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.

翻译：我们研究如何在有限区间上构造一个具有给定“粗糙度”且边际分布联合矩有限随机过程。首先，我们沿一般划分序列扩展Ciesielski同构，并基于Schauder系数给出函数Hölder正则性的刻画。利用此刻画，我们提出一个更优的（路径wise）Hölder指数估计量。作为另一个应用，我们构造了具有与（分数）布朗运动相同的某些路径性质及边际分布有限矩的伪（分数）布朗运动。这些过程属于非高斯随机过程族，在统计上难以与真实的（分数）布朗运动区分。