Hybrid Switching Diffusions : Properties and Applications

Hybrid switching diffusions, also known as switching diffusions or regimeswitching systems, have drawn a signifi cant amount of attention in recent decades in a variety of applications due to their ability to model complex systems with uncertainty. Control and systems engineers have long been familiar with the fundamental idea of using a continuous stochastic differential equation to describe a dynamic system with random disturbances. However, in the real world the confi guration of a complex dynamic system can undergo sudden and signifi cant changes due to changes in the environment. Impacts of such discrete events on modeling need to be taken into account to better characterize many realistic systems, and this need has motivated the emergence and growth of the study of regime-switching systems. A common setup for a switching diffusion is to use several continuous stochastic differential equations, each for a specific regime, together with a Markov chain to model the random switching among regimes. Consequently, both continuous dynamics and discrete events are present in the switching diffusion formulation, providing better models for applications such as stochastic control and optimization, estimation and filtering, manufacturing systems, operations management, queueing networks, financial mathematics, and financial engineering. Many questions have arisen regarding hybrid switching diffusions, and we can roughly divide them into three groups. The first group is concerned with fundamental properties of the stochastic processes governed by switching diffusions. Issues such as the existence and uniqueness of solutions, boundedness and stability, regularity, recurrence, and ergodicity have been addressed. These properties have been long established for continuous diffusions without switching. A natural question to ask is: Under what conditions do these properties hold for switching diffusions? Research on these topics is the subject of probability theory and stochastic analysis. The second group of questions deals mainly with approximate solutions and numerical algorithms for switching diffusions. Due to the coexistence of both continuous dynamics and discrete events and their random interactions in regimeswitching systems, closed-form solutions are possible only for a few simple cases. Thus, seeking effective and efficient approximate and numerical solutions has been the subject of many research papers. Convergence of the approximate methods and rate of convergence are relevant as well. The third group involves applications. In financial engineering, for example, it has been acknowledged that a regimeswitching model can effectively describe the well-known volatility smile phenomena. Many efforts have been made in tackling the problems of option pricing and hedging, optimal portfolio selections, and optimal investment decisions in the switching diffusion setup. The book Hybrid Switching Diffusions provides a remarkable coverage of up-to-date, cutting-edge research results on switching diffusions. While many works treat the Markov chain for discrete events as an independent process of the continuous state variable, this book moves one step ahead by allowing the discrete process to depend on the continuous state process, which certainly provides more flexibility for modeling real-world systems. Both theory and applications are covered in this monograph. This book addresses the aforementioned three questions. The material presented in the book is divided into four parts. Parts 1 and 3 are concerned with the first question, Part 2 the second, and Part 4 the third, respectively. Part 1 of the book, composed of three chapters, deals with basic properties of switching diffusions. In Chapter 2 the authors present a definition of the switching diffusion processes to be studied. They then discuss the regularity, weak continuity, Feller and strong Feller properties, and I EEE Control Systems Magazine welcomes suggestions for books to be reviewed in this column. Please contact either Michael Polis or Zongli Lin, associate editors for book reviews.