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From \(N\) parameter fractional Brownian motions to \(N\) parameter multifractional Brownian motions. (English) Zbl 1135.60020
Summary: Multifractional Brownian motion is an extension of the well-known fractional Brownian motion where the Hölder regularity is allowed to vary along the paths. In this paper, two kinds 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 Hölder 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.

60G15 Gaussian processes
60G05 Foundations of stochastic processes
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