This book provides recent results on the stochastic approximation of systems by weak convergence techniques. General and particular schemes of proofs for average, diffusion, and Poisson approximations of stochastic systems are presented, allowing one to simplify complex systems and obtain numerically tractable models.

The systems discussed in the book include stochastic additive functionals, dynamical systems, stochastic integral functionals, increment processes and impulsive processes. All these systems are switched by Markov and semi-Markov processes whose phase space is considered in asymptotic split and merging schemes. Most of the results from semi-Markov processes are new and presented for the first time in this book.

Contents:

• Markov and Semi-Markov Processes
• Stochastic Systems with Switching
• Stochastic Systems in the Series Scheme
• Stochastic Systems with Split and Merging
• Phase Merging Principles
• Weak Convergence
• Poisson Approximation
• Applications
• Appendices:
• Weak Convergence of Probability Measures
• Some Limit Theorems for Stochastic Processes
• Some Auxiliary Results

“Introductory facts related to weak convergence of the stochastic process and the convergence of semimartigales, theorems, generalizations for results previously published by the authors, proofs and applications are well organized and distributed throughout nine chapters and three appendices.”

“This book may serve as a textbook for graduate students, postdoctoral seminars or courses for applied scientists and engineers in stochastic approximation of complex systems: queueing, reliability, risk, finance.”