Probabilistic and Statistical Methods of Decomposition of the Volatility of Chaotic Processes

The book describes probabilistic mathematical models of chaotic processes and methods for their statistical analysis. 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 authors use an asymptotic approach 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. This approach leads to the distributions of increments of the processes on time macro-scales havivng the form of mixtures of normal distributions and makes it possible not only to find formal probabilistic models of chaotic stochastic processes, but also to give a reasonable theoretical explanation of their adequacy based on mild assumptions concerning the inner structure of the characteristics under investigation. The book proposes a new multivariate interpretation of the volatility of these processes by representing the distributions (logarithms) of increments of financial index evolution processes or those of plasma turbulence as mixtures of normal distribution. For the statistical analysis of chaotic processes it proposes a method of moving separation of mixtures (MSM method) enabling spontaneous decomposition of process volatility into dynamic and diffusive components. Much attention is paid to the analytical and asymptotic properties of mixtures of normal distributions. Systematic consideration is given to statistical procedures for such numerical separation of mixtures as the EM algorithm and its modifications, grid separation methods. The author discusses questions of optimal implementation of these methods and considers examples of CPC method application to analyze the impact of information interventions in financial markets and the data obtained in plasma turbulence experiments.

For postgraduate students and university professors who are interested in the modern state of research in the field of probabilistic and statistical modeling of chaotic stochastic processes as well as scientists, engineers, specialists in the application of mathematical methods and applied statistics for analyzing characteristics of financial markets and plasma turbulence.