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.

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 is no longer adequate.