The book provides a systematic presentation of the theory of general stochastic systems. The results are based on the research performed at the Institute for Informatic Problems of the Russian Academy of Sciences. The rough overview is as follows: Chapter 1 introduces the concept of (deterministic or stochastic) systems, Chapters 2, 3 and 4 give the probability theory and stochastic analysis background, and Chapters 5, 6 and 7 the general theory of stochastic systems, and methods of linear and nonlinear stochastic systems theory and their applications. Each chapter has a section of problems at the end, all in all over 500, without solutions. The general theory of stochastic systems developed here is based on the finite-dimensional characteristics of the solution processes. This approach is suitable for engineers and scientists working in applied fields.
In some more detail the large amount of material presented is the following. Chapter 1 introduces the concept of (deterministic or stochastic) systems, i.e. mathematical models consisting of the space of inputs, the space of outputs, the state space and the functional relations connecting the inputs, the outputs and the state vector of the system. The latter describe the behaviour of the system, they may be deterministic or stochastic differential or difference equations, linear or nonlinear. The concepts of transfer functions and frequency response functions for the systems considered are also introduced. Examples from engineering are given to illustrate these concepts.
Chapter 2 provides an introduction to probability spaces, random variables, conditional probabilities and stochastic processes and their measurability, continuity, separability and differentiability. The first half of Chapter 3 deals with expectation and other moments, conditional moments, characteristic functions and functionals. In the second half sequences of probability measures, distributions in Banach spaces, normal distributions in linear spaces, the approximate representation of distributions and canonical expansions are considered. Chapter 4 presents mean-square analysis, i.e. limits, continuity, differentiation and integration, and the stochastic integral. In addition, generalized random functions, such as white noise, and integral canonical and spectral representations are introduced.
Chapter 5 treats the general theory of stochastic systems. Stochastic differential equations with a Wiener process and a Poisson measure as driving processes are introduced, several versions of Itô-formulas and an existence and uniqueness result for solutions are given. Random differential equations, i.e. differential equations depending on a random parameter, are also considered. The central problem of the theory of continuous stochastic systems is formulated as the probability analysis of the multi-dimensional distributions of the processes satisfying stochastic differential equations. A section on numerical methods for stochastic differential equations follows. Then the determination of one-dimensional and multi-dimensional characteristic functions and densities of the solutions of stochastic differential equations are considered. For example, the Fokker-Planck-Kolmogorov equations are derived.
Chapters 6 and 7 are devoted to special methods for calculating moments and their derivatives, and characteristic functions of solutions of linear (in Chapter 6) and nonlinear (in Chapter 7) stochastic integral, differential, difference and other operator equations in finite- or infinite-dimensional spaces. Several methods for approximating the multi-dimensional distributions of the solutions are presented. The calculations are given in detail. A large amount of applications, such as the accuracy analysis of single loop nonlinear control systems or the normalization of queueing systems, conclude the book.