2005/03/09 by E. Herbin, Érick Herbin, Herbin, E.
Economics, Econometrics and Finance · Mathematics · #60G15 #60G17 #60G18 #62G05 #Complex Systems and Time Series Analysis #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #math.PR #msc:60G15 #msc:60G17 #msc:60G18 #msc:62G05
paper · pdf · doi:10.48550/arxiv.math/0503182
36 pages
arxiv created 2005/03/09 · openalex publication_date 2005/03/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Multifractional Brownian motion is an extension of the well-known fractional Brownian motion where the Holder regularity is allowed to vary along the paths. In this paper, two kind of multi-parameter extensions of mBm are studied: one is isotropic while the other is not. For each of these processes, a moving average representation, a harmonizable representation, and the covariance structure are given. The Holder regularity is then studied. In particular, the case of an irregular exponent function H is investigated. In this situation, the almost sure pointwise and local Holder exponents of the multi-parameter mBm are proved to be equal to the correspondent exponents of H. Eventually, a local asymptotic self-similarity property is proved. The limit process can be another process than fBm.