2011/09/07 by Antoine Ayache, Ayache, Antoine · 1 citation
Economics, Econometrics and Finance · Engineering · Mathematics · #60G15 #60G17 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications #advanced mathematical theories #math.PR #msc:60G15 #msc:60G17
paper · pdf · doi:10.48550/arxiv.1109.1617
openalex publication_date 2011/09/07 · arxiv created 2012/02/18 · arxiv updated 2012/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be an arbitrary centered Gaussian process whose trajectories are, with probability one, continuous nowhere differentiable functions. It follows from a classical result, derived from zero-one law, that, with probability one, the trajectories of X have the same global Hölder regularity over any compact interval, that is the uniform Hölder exponent does not depend on the choice of a trajectory. A similar phenomenon happens with their local Hölder regularity measured through the local Hölder exponent. Therefore, it seems natural to ask the following question: does such a phenomenon also occur with their pointwise Hölder regularity measured through the pointwise Hölder exponent? In this article, using the framework of multifractional processes, we construct a family of counterexamples showing that the answer to this question is not always positive.