PREFACEThe monograph is dedicated to the systematic presentation of the applied advanced theory and analytical methods for the stochastic systems, i.e. the dynamical systems described by the finite-and infinitedimensional stochastic differential, difference, integral, integrodifferential etc. equations. It is based on the results of the fundamental research performed in the Institute for Informatic Problems of the Russian Academy of Sciences in the context of the scientific program "Stochastic Systems" and the lecture courses delivered by the authors in the domestic and the foreign Technical Universities.The book may be used as the textbook for the applied mathematics faculties of the universities. The unified procedure, thorough selection of the examples and the problems (over 500) and the large applications make this book useful for the graduates, the post-graduates and the lecturers. The book is of considerable interest also for the mathematicians who deal with the stochastic equations and their applications.The construction of general stochastic systems theory is based on the equations for the multi-dimensional characteristic functions and functionals. The stochastic differential equations with the arbitrary processes with the independent increments are studied. The equations with the Wiener and the Poisson processes are considered as a special case. The methods of the parametrization of the multi-dimensional distributions in the nonlinear stochastic systems based on the moments, the semiinvariants, the quasimoments and on the consistent orthogonal expansions are stated systematically. Special attention is paid to linear and reducible to linear stochastic systems theory based on the canonical representations (canonical expansions and integral canonical representations). Most attention has been concentrated on the structural theory of the composed stochastic systems on the grounds of the conditional probability measures.Chapter 1 is devoted to the mathematical models of the dynamic systems under conditions of the random disturbances and their characteristics. The linear and the nonlinear continuous, discrete and continuousdiscrete systems described by the stochastic equations in the finite-and the infinite-dimensional spaces are considered. Special attention is paid to the composed stochastic systems.The main notions of probability distributions theory of the random variables, the random processes and the random functions are stated in Chapter 2. After determining the probability spaces, the conditional VI Preface probabilities, probabilities in the finite and the infinite products of the spaces are considered. The conditions of the existence of the regular probabilities are established. Further the different probability measures of the random functions and the probabilities of events connected with the random functions are studied in detail. The last sections are devoted to the distributions of the separable random functions, to the derivation of the criteria of the continuity and the differentiability of th...

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Theoretical propositions of new probabilistic methodology of analysis, modeling, estimation and control in stochastic organizational-technical-economic systems (OTES) based on stochastic CALS informational technologies are considered. Stochastic integrated logistic support (ILS) of OTES modeling life cycle (LC), stochastic optimal of current state estimation in stochastic media defined by internal and external noises (including specially organized OTES-NS (noise support) and stochastic OTES optimal control) according to social-technical-economic-support criteria in real time by informational-analytical tools (IAT) of global type are presented. OTES-CALS are nonlinear and continuous-discrete. So we use approximate methods of normal approximation of probabilistic densities both for modeling and estimation. Spectrum of possibilities may be broaden by solving problems of OTES-CALS integration for existing markets of finances, goods and services. Analytical modeling, analysis, parametric optimization and optimal stochastic processes regulation in limits of illustrate some technologies and IAT given plans.

The methods of the control stochastic systems (CStS) research based on the parametrization of the distributions permit to design practically simple software tools. These methods give the rapid increase of the number of equations for the moments, the semiinvariants, coefficients of the truncated orthogonal expansions of the state vector Y, and the maximal order of the moments involved. For structural parametrization of the probability (normalized and nonnormalized) densities, we shall apply the ellipsoidal densities. A normal distribution has an ellipsoidal structure. The distinctive characteristics of such distributions consist in the fact that their densities are the functions of positively determined quadratic form of the centered state vector. Ellipsoidal approximation method (EAM) cardinally reduces the number of parameters. For ellipsoidal linearization method (ELM), the number of equations coincides with normal approximation method (NAM). The development of EAM (ELM) for CStS analysis and CStS filtering are considered. Based on nonnormalized densities, new types of filters are designed. The theory of ellipsoidal Pugachev conditionally optimal control is presented. Basic applications are considered.

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