We study how to construct a stochastic process on a finite interval with given `roughness' and finite joint moments of marginal distributions. Our construction method is based on Schauder representation along a general sequence of partitions and has two ramifications. We show that the variation index of a process along a given partition sequence (the infimum value $p$ such that the $p$-th variation is finite) may not be equal to the reciprocal of H\"older exponent, and provide a pathwise estimator of H\"older exponent. Moreover, we construct a non-Gaussian family of stochastic processes which are statistically indistinguishable from (fractional) Brownian motions. Therefore, when observing a sample path from a process in a financial market such as a price or volatility process, we should not measure its H\"older roughness by computing $p$-th variation and should not conclude that the sample is from Brownian motion or fractional Brownian motion even though it exhibits the same properties of those Gaussian processes.

翻译：我们研究如何在有限区间上构造具有给定“粗糙度”且边缘分布有限联合矩的随机过程。我们的构造方法基于沿一般分割序列的肖德尔表示，并衍生出两个分支。我们证明，沿给定分割序列的过程变差指数（使$p$次变差有限的下确界值$p$）可能不等于赫尔德指数的倒数，并给出了赫尔德指数的路径估计量。此外，我们构造了一族与（分数）布朗运动在统计上不可区分的非高斯随机过程。因此，当从金融市场中的过程（如价格或波动率过程）观测样本路径时，我们不应通过计算$p$次变差来衡量其赫尔德粗糙度，也不应仅因样本表现出与这些高斯过程相同的性质就断定其来自布朗运动或分数布朗运动。