Applications of Schauder-type basis: estimating Hölder exponent, fake fractional Brownian motion

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.