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The notions of self-similarity, scaling, fractional processes and long range dependence have been repeatedly used to describe properties of financial time series: stock prices, foreign exchange rates, market indices and commodity prices. We discuss the relevance of these properties in the context of financial modelling, their relation with the basic principles of financial theory and possible economic explanations for their presence in financial time series.

The multiscale fractional Brownian motion provides an example of process with long memory which is arbitrage free. After having recalled the definition of the long memory and the notion of arbitrage free price process, we derive the price of European options. This one is the Black-Scholes price using the ”high-frequency” volatility parameter. This shows that the presence or absence of long memory is not a relevant question for pricing European options.

We give an overview of limit results for a selection of stochastic models that involve heavy-tailed distributions, exhibit long-range dependence and are naturally parametrized by a tail-index. Under aggregation of independent subsystems and simultaneous time or space rescaling, the asymptotic behavior of such systems varies considerably. It is the relative speed of aggregation degree and rescaling that determines the nature of the limit process, ranging from fractional Brownian motion to stable Lévy processes. The limits obtained for a generalized example based on a spatial Poisson grains model include a fractional Brownian field. We are particularly interested in the intermediate scaling regime, bridging Gaussian and stable asymptotics. One feature shared by all limit processes is identified as an aggregatesimilarity property.

We develop a non-parametric statistical test for self-similarity based on the crossing tree and use simulation experiments to test its performance. It is applied to a number of packet traces both to determine the range of scales over which they appear self-similar and to detect temporal changes in the mean packet arrival rate and/or scaling behaviour.

Let be a contraction mapping on an appropriate Banach space (). Then the evolution equation = − can be used to produce a continuous evolution () from an arbitrary initial condition ∊ () to the fixed point ∊ () of . This simple observation is applied in the context of iterated function systems (IFS). In particular, we consider (1) the Markov operator (on a space of probability measures) associated with an -map IFS with probabilities (IFSP) and (2) the fractal transform (on functions in (), for example) associated with an -map IFS with greyscale maps (IFSM), which is generally used to perform fractal image coding. In all cases, the evolution equation takes the form of a nonlocal differential equation.

Such an evolution equation technique can also be applied to complex analytic mappings which are not strictly contractive but which possess invariant attractor sets. A few simple cases are discussed, including Newton's method in the complex plane.

Dealing with several structures of different complexities, we adopt various strategies to extract information. The general idea is to find appropriate tools to analyze the variation of the corresponding autocorrelation functions. First, for homogeneous media under different conditions, we recover, in a statistical way, a relationship between porosity and the autocorrelation function. Then, for low-complexity textures, we exploit this relationship to extract complementary parameters from the autocorrelation function beyond porosity using spectral analysis. For fractal-like structures, we process them according to their porosity. For fat fractals, usually used as synthetic models of porous media, we combine the regularization dimension, a method proposed to estimate the curve variation, with the autocorrelation function. This leads to a more robust classification. For fractals of negligible porosity, such as fractals of non-integer dimension, we discuss how the method HMSF we developed serves as an original means to estimate the Hausdorff dimension and how it can be exploited to give complementary characteristic parameters.

An analytical approach to fractal inverse problem is presented in this paper. We recall the construction of an Hilbert space with address functions, that constitute a general framework for fractal modeling, since it includes IFS made fractals. A large scale of applications is shown, ranging from scalar function approximation to image compression.