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的随机过程。首先，我们沿一般分割序列扩展了Ciesielski同构，并给出了函数赫尔德正则性基于其Schauder系数的刻画。利用这一刻画，我们提出了一种更优的（路径-wise）赫尔德指数估计量。作为附加应用，我们构造了具有与（分数）布朗运动相同路径性质及边际分布有限矩的假（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（分数）布朗运动区分。