Abstract:
Probabilistic mathematical models of chaotic processes and methods for their statistical analysis are described. The focus is put on the special class of mathematical models of stochastic chaotic processes – subordinated Wiener processes (processes of Brownian motion with random time). To explain the choice of these models the asymptotic approach is used. This approach is based on limit theorems for compound doubly stochastic Poisson processes (compound Cox processes) which can be regarded as the best mathematical models of non-homogeneous (or non-stationary) chaotic flows on time micro-scales. It leads to the fact that the distributions of increments of these processes on time macro-scales have the form of mixtures of normal distributions and makes it possible not only to find formal stochastic models of chaotic processes, but also to give a reasonable theoretical explanation of their adequacy based on mild assumptions concerning the inner structure of the studied characteristics. A new multivariate interpretation of the volatility of these processes is proposed resulting from that the distributions of (the logarithms) of increments of the processes of the evolution of financial indexes or the processes of plasma turbulence are mixtures of normal distribution. For the statistical analysis of chaotic processes a method of moving separation of mixtures (MSM-method) is proposed which allows to spontaneously decompose the volatility of the process in dynamic and diffusive components. Statistical procedures of separation of mixtures (EM-algorithm and its modifications), as well as new grid methods of separation of mixtures are discussed as well as the issues of effective usage of these methods. Examples of application of the MSM-method for the separation of mixtures of distributions to the analysis of informational interventions at the financial markets and to the analysis of plasma turbulence experimental data are presented.
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