Self-similarity in random collision processes.

Multifractional Processes with Random Exponent

Multifractional Processes with Random Exponent (MPRE) are obtained by replacing the Hurst parameter of Fractional Brownian Motion (FBM) with a stochastic process. This process need not be independent of the white noise generating the FBM. MPREs can be conveniently represented as random wavelet series. We will use this type of representation to study their Hölder regularity and their self-simila...

The genealogy of self-similar fragmentations with negative index as a continuum random tree

We encode a certain class of stochastic fragmentation processes, namely self-similar fragmentation processes with a negative index of self-similarity, into a metric family tree which belongs to the family of Continuum Random Trees of Aldous. When the splitting times of the fragmentation are dense near 0, the tree can in turn be encoded into a continuous height function, just as the Brownian Con...

Remarks on collision manifolds and nonexistence of non self-similar collision solutions in the 3-vortex problem

Statistical Self-Similarity of One-Dimensional Growth Processes

For one-dimensional growth processes we consider the distribution of the height above a given point of the substrate and study its scale invariance in the limit of large times. We argue that for self-similar growth from a single seed the universal distribution is the Tracy-Widom distribution from the theory of random matrices and that for growth from a flat substrate it is some other, only nume...

Fractals or I.I.D.: Evidence of Long-Range Dependence and Heavy Tailedness from Modeling German Equity Market Volatility

Several studies find that return volatility of stocks tends to exhibit long-range dependence, heavy tailedness, and clustering. In this study, we use high-frequency data to empirically investigate whether a sample of stocks exhibit those characteristics. Because we do find those characteristics, as suggested by Rachev and Mittnik (2000) we employ self-similar processes to capture them in modeli...