Fractal Dimensions for Fractal Structures and their Applications to Financial Markets

Area 01 – Scienze matematiche e informatiche

This book provides a new approach to fractal dimension theory from the point of view of Asymmetric Topology through fractal structures. Fractal dimension is the main invariant of each fractal set which gives information about the irregularities that it presents when being examined with enough level of detail. In this work, the authors develop new models to calculate the fractal dimension for any subspace with respect to a fractal structure which generalize the classical fractal dimension definitions, namely, both the Hausdorff dimension and the box-counting dimension. They also include some specific results for self-similar sets. In addition, the new definitions of fractal dimension can be calculated in empirical applications unlike it happens with the Hausdorff dimension, which may result hard or even impossible to calculate. The second part of the book contains an interesting application of fractal dimension for fractal structures to financial markets. The authors introduce some tools and prove some results which connect the fractal dimension with the Hurst exponent which has been classically used to look for long-range dependence in financial time series. These theoretical results allow to provide new algorithms especially accurate to estimate the self-similarity exponent of a wide range of processes which includes fractional Brownian motions and Lévy stable motions as particular cases. Moreover, some specific analysis have also been carried out for real stocks and international indexes.

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