Stochastic Systems

Abstract

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...

Optimal Filtering for Linear States over Polynomial Observations

“…Using the Ito formula (see (31)) for the stochastic differential of the nonlinear function h(x, t), where x(t) satisfies the equation (1), the following equation is obtained for z(t)…”

“…Hence, the optimal filter for the polynomial system states with unmeasured linear part and polynomial multiplicative noise over linear observations, obtained in (29), can be applied to solving this problem. Indeed, as follows from the general optimal filtering theory (see (31)), the optimal filtering equations take the following particular form for the system (5), (1), (6) …”

“…, which should be calculated to obtain a closed system of filtering equations (see (31)), would also include only polynomial terms of x. Then, those polynomial terms can be represented as functions of m(t) and P(t) using the following property of Gaussian random variable [z(t), x(t)] − m(t): all its odd conditional moments,…”

“…Designing the optimal filter over polynomial observations presents a significant advantage in the filtering theory and practice, since it enables one to address some filtering problems with observation nonlinearities, such as the optimal cubic sensor problem (30). The optimal filtering problem is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance (31). As the first result, the Ito differentials for the optimal estimate and error variance corresponding to the stated filtering problem are derived.…”

Analysis of Nonlinear Stochastic Systems with Jumps Generated by Erlang Flow of Events

“…These systems are called the jump-diffusion systems or the stochastic systems with random quantization period. Jumps may have different characteristics that describe intervals between them and their amplitudes [1,2].…”