Multifractional Brownian motion (mBm) was introduced to overcome certain limitations of the classical fractional Brownian motion (fBm). The major di erence between the two processes is that, contrarily to fBm, the almost sure H older exponent of mBm is allowed to vary along the trajectory, a useful feature when one needs to model processes whose regularity evolves in time, such as Internet tra c or images. Various properties of mBm have already been investigated in the literature, related for instance to its dimensions or the statistical estimation of its pointwise H older regularity. However, the covariance structure of mBm has not been investigated so far. We present in this work an explicit formula for this covariance. Since mBm is a zero mean Gaussian process, such a formula provides a full characterization of its stochastic properties. We brie y report on some applications, including the synthesis problem and the long term structure : in particular, we show that the increments of mBm exhibit long range dependence under general conditions.
We present a general method for constructing stochastic processes with prescribed local form. Such processes include variable amplitude multifractional Brownian motion, multifractional α-stable processes, and multistable processes, that is processes that are locally α(t)-stable but where the stability index α(t) varies with t. In particular we construct multifractional multistable processes, where both the local self-similarity and stability indices vary.
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