Evolution stochastic systems. Averaging algorithms and diffusion approximation. (Q2701756)

Theory of evolution stochastic systems (ESS) has many different aspects. If such systems are investigated by a mathematician-probabilist, then he is interested in the averaging and diffusion approximation w.r.t. the properties of random media (r.m.) (ergodicity, reducibility, etc.). Specialists in mechanics are interested in their stability. Physicists may be interested in the probabilistic representation of solutions to evolutionary equations in random media (for example, Schrödinger equation for random potentials, Klein-Gordon equation for random plasma frequencies, etc.). NEWLINENEWLINENEWLINERandom evolutions (RE) are an abstract mathematical model of an ESS in random media, and RE are described by stochastic operator integral equations in a Banach space. The investigation of RE is usually carried out by using the asymptotic perturbation theory that is based on the asymptotic analysis of Markov renewal equations for expectations of RE and also on the theory of operators with perturbed spectrum. With the help of these methods one can obtain algorithms for the averaging and diffusion approximation for RE, but especially an effective approach to the investigation of RE turns out to be martingale one. It allows to obtain a solution to a martingale problem, rates of convergence in the limit theorems for RE, stability of RE, etc. Since ESS in r.m. are realizations of RE, then the results for RE are transferred to ESS, which have certain methodological preferences. NEWLINENEWLINENEWLINEUp to this date there were six books on RE: 1) \textit{M. A. Pinsky} [Lectures on random evolution (Singapore, 1991)]; 2) the authors [Semi-Markov random evolutions (Kiev, 1992)]; 3) the authors [ Semi-Markov random evolutions (1995; Zbl 0813.60083)]; 4) the authors [Evolution of systems in random media (Boca Raton, 1995)]; 5) \textit{A. Swishchuk} [Random evolutions and their applications (1997; Zbl 0892.60089)]; 6) \textit{A. V. Swishchuk} [Random evolutions and their applications. New trends (Dordrecht, 2000)]. NEWLINENEWLINENEWLINEThe first book contains an axiomatic approach to Markov RE and their applications to kinetic theory, isotropic transport theory on manifolds and stability of random oscillator. In the second book a general theory of semi-Markov RE (SMRE), limit averaging theorems and diffusion approximation for SMRE and their applications to some stochastic systems are described. The third book is the translation of the second one. The range of martingale and semigroup techniques available to RE run in the fourth book through the whole spectrum, from averaging, diffusion approximation, normal deviations, stability, control and rates of convergence, to martingale problems and stochastic integral equations in Banach space. The fifth book is a handbook on RE and their applications, and contains some results in stochastic calculus for integrals over martingale measures with applications to RE integral equations, and hedging of options with semi-Markov volatility. The sixth book contains new results in RE and their applications in financial and insurance mathematics stochastic models, stability and control of RE and their applications to financial and insurance stochastic models and statistics of financial models. NEWLINENEWLINENEWLINEBut there is a shortage of books where concrete methods and algorithms allowing to obtain both qualitative and quantitative results for ESS are described. Especially books with the help of which it is possible sufficiently quick to learn a constructive part of the theory of RE by examples of ESS in r.m. Namely, methods of calculations of averaging and diffusion approximation, estimations of values of deviations, etc. NEWLINENEWLINENEWLINEThe main contents of this book are concrete methods and algorithms allowing to find solutions to problems of the theory of RE. For demonstration of their effectivity the most typical ESS in r.m. are considered: traffic, storage, branching processes, etc. The contents of the book is the following: Chapter 1 is devoted to the description of r.m. by Markov renewal processes and semi-Markov processes. Chapter 2 deals with algorithms of averaging of semi-Markov processes, martingale characterization and compactness in series scheme. MRE and ESS in random media are described in Chapter 3. Algorithms of averaging for RE and ESS, and rates of closeness of real and averaged systems are studied in Chapter 4. Algorithms of diffusion approximation for RE and ESS with balance condition, and rates of closeness of real and averaged systems in diffusion approximation are investigated in Chapter 5. Normal deviations of initial RE and ESS from averaged, when balance condition is not fulfilled, are described in Chapter 6. Martingale methods of investigation of ESS by stochastic integral functionals and diffusion processes with semi-Markov switchings are studied in Chapter 7. NEWLINENEWLINENEWLINEThis book will be useful for those who are interested in applications of RE and ESS and want to use these algorithms for practical applications.