Synopses & Reviews
This book presents a comprehensive study of hybrid switching diffusion processes and their applications. The motivations for studying such processes originate from emerging and existing applications in wireless communications, signal processing, queueing networks, production planning, biological systems, ecosystems, financial engineering, and modeling, analysis, and control and optimization of large-scale systems, under the influence of random environment. One of the distinct features of the processes under consideration is the coexistence of continuous dynamics and discrete events. This book is written for applied mathematicians, applied probabilists, systems engineers, control scientists, operations researchers, and financial analysts. Selected materials from the book may also be used in a graduate level course on stochastic processes and applications or a course on hybrid systems. A large part of the book is concerned with the discrete event process depending on the continuous dynamics. In addition to the existence and uniqueness of solutions of switching diffusion equations, regularity, Feller and strong Feller properties, continuous and smooth dependence on initial data, recurrence, ergodicity, invariant measures, and stability are dealt with. Numerical methods for solutions of switching diffusions are developed; algorithms for approximation to invariant measures are investigated. Two-time-scale models are also examined. The results presented in the book are useful to researchers and practitioners who need to use stochastic models to deal with hybrid stochastic systems, and to treat real-world problems when continuous dynamics and discrete events are intertwined, in which the traditional approach using stochastic differential equations alone
Review
From the reviews: "The book Hybrid Switching Diffusions provides a remarkable coverage of up-to-date, cutting-edge research results on switching diffusions. ... As I read through the book, I was impressed by the vast number of topics covered as well as the level of technical details. ... provides a thorough, up-to-date development of regime-switching diffusions. It provides a valuable reference for applied mathematicians and control scientists. The book can also be useful to researchers in other application fields who seek suitable mathematical models that may fit their own systems." (Ruihua Liu, IEEE Control Systems Magazine, Vol. 30, October, 2010) "This book is written for scientists, engineers, and financial analysts interested in processes described by the coexistence of discrete events and continuous dynamics. Among the topics treated are existence and uniqueness of solutions of switching diffusion equations, regularity, well posedness, recurrence, ergodicity, stability, numerical methods, and two-time-scale processes." (IEEE Control Systems Magazine, Vol. 30, June, 2010)
Review
From the reviews:
"The book Hybrid Switching Diffusions provides a remarkable coverage of up-to-date, cutting-edge research results on switching diffusions. ... As I read through the book, I was impressed by the vast number of topics covered as well as the level of technical details. ... provides a thorough, up-to-date development of regime-switching diffusions. It provides a valuable reference for applied mathematicians and control scientists. The book can also be useful to researchers in other application fields who seek suitable mathematical models that may fit their own systems." (Ruihua Liu, IEEE Control Systems Magazine, Vol. 30, October, 2010)
"This book is written for scientists, engineers, and financial analysts interested in processes described by the coexistence of discrete events and continuous dynamics. Among the topics treated are existence and uniqueness of solutions of switching diffusion equations, regularity, well posedness, recurrence, ergodicity, stability, numerical methods, and two-time-scale processes." (IEEE Control Systems Magazine, Vol. 30, June, 2010)
Synopsis
This book focuses on switching diffusion processes involving both continuous dynamics and discrete events. The first part, including three chapters, presents basic properties such as Feller and strong Feller, recurrence, and ergodicity. With a brief review of existence and uniqueness of solutions of switching diffusions, basic properties such as recurrence, Feller properties etc. are dealt with. The second part of the book is devoted to numerical solutions of switching diffusions. Containing three chapters, the third part focuses on stability. Chapter seven and chapter eight proceed with the stability analysis. The approach is based on Liapunov function methods. For convenient references, an appendix including a number of mathematical preliminaries are placed at the end of the book. Topics discussed here including Markov chains, martingales, Gaussian processes, diffusions, jump diffusions, and weak convergence methods. Although detailed developments are often omitted, appropriate references are provided for the reader for further reading.
Synopsis
and Motivation.- Basic Properties, Recurrence, Ergodicity.- Switching Diffusion.- Recurrence.- Ergodicity.- Numerical Solutions and Approximation.- Numerical Approximation.- Numerical Approximation to Invariant Measures.- Stability.- Stability.- Stability of Switching ODEs.- Invariance Principles.- Two-time-scale Modeling and Applications.- Positive Recurrence: Weakly Connected Ergodic Classes.- Stochastic Volatility Using Regime-Switching Diffusions.- Two-Time-Scale Switching Jump Diffusions.
Synopsis
This book encompasses the study of hybrid switching di usion processes and their applications. The word \hybrid signi es the coexistence of c- tinuous dynamics and discrete events, which is one of the distinct features of the processes under consideration. Much of the book is concerned with the interactions of the continuous dynamics and the discrete events. Our motivations for studying such processes originate from emerging and - isting applications in wireless communications, signal processing, queueing networks, production planning, biological systems, ecosystems, nancial engineering, and modeling, analysis, and control and optimization of lar- scale systems, under the in uence of random environments. Displaying mixture distributions, switching di usions may be described by the associated operators or by systems of stochastic di erential eq- tions together with the probability transition laws of the switching actions. We either have Markov-modulated switching di usions or processes with continuous state-dependent switching. The latter turns out to be much more challenging to deal with. Viewing the hybrid di usions as a number of di usions joined together by the switching process, they may be se- ingly not much di erent from their di usion counterpart. Nevertheless, the underlying problems become more di cult to handle, especially when the switching processes depend on continuous states. The di culty is due to the interaction of the discrete and continuous processes and the tangled and hybrid information pattern.
Synopsis
Focusing on switching diffusion processes that involve both continuous dynamics and discrete events, this comprehensive study moves from basic properties such as Feller and strong Feller to the numerical solutions of switching diffusions and stability.
Table of Contents
Preface.-Introduction and Motivation.-Switching Diffusion.- Recurrence.-Ergodicity.-Numerical Approximation.-Numerical Approximation to Invariant Measures.-Stability.-Stability of Switching ODE.-Invariance Principles.-Positive Recurrence: Multi-ergodic-class of Switching Processes.-Stochastic Volatility Using Regime-switching Diffusions.- Two-time-scale Switching Jump-diffusions.-Appendix.-References.