Markov-Modulated Brownian Motion

This book explores regime-switching Brownian motion, a class of stochastic processes widely used in fields such as mathematical finance, risk theory, queueing theory, and epidemiological modeling. These processes are studied within the Markovian regime-switching framework, which captures dynamic environments characterized by shifts between different states or "regimes"for example, economic cycles, seasonal environmental variations, or short-term surges in activity.

The matrix-analytic approach, introduced approximately fifty years ago in the context of classical queueing theory, serves as the foundation for this analysis. This methodology emphasizes the examination of process trajectories over time, drawing insights from the interplay between analytic derivations and their physical or probabilistic interpretations. A central objective of the matrix-analytic framework is to produce solutions that are not only analytically tractable but also amenable to efficient, stable numerical algorithmsfacilitating practical implementation using standard computational tools. This enables both quantitative performance evaluation and qualitative system understanding.

Originally developed for telecommunication network modeling, matrix-analytic methods have since found applications across a broad spectrum of disciplines, including risk analysis, branching processes, and epidemiology.

This book is the first to offer a systematic application of matrix-analytic techniques to Markov-modulated Brownian motion, filling a gap in the literature and providing a valuable resource for researchers and practitioners alike.

The intended audience includes specialists in stochastic processes and their applicationssuch as applied probabilists, actuaries, financial analysts, systems and operations researchers, applied statisticians, and engineers in telecommunications and electrical domains. Readers are expected to have a background in advanced undergraduate calculus, linear algebra, and introductory stochastic processes.

The book systematically applies the Matrix-Analytic approach to the resolution of Markov-modulated models It emphasizes the interplay between algebraic manipulations and the physical behaviour of the stochastic process It gives two simple, provably efficient and stable numerical algorithms to compute any of the fundamental matrices

Autorentext

Guy Latouche received his Ph.D from the Université libre de Bruxelles in 1976. He is professor emeritus from the Université libre de Bruxelles where he taught classes on stochastic processes and their applications, computer programming, management information systems, and formal methods for proofs of programs. He has been a visiting professor at the University of Delaware, visiting professor at the Tokyo Institute of Technology, and a frequent short-term visitor to the Universities of Adelaide, of Melbourne and of Pisa. His research interests include various aspects of applied probability: matrix methods in Markov models, traffic models for telecommunication systems, and nearly completely decomposable systems. He has contributed extensively to the development of computational methods for the analysis of Markov models and he is internationally acknowledged as one of the world leaders in the field. He is co-author of three books, co-editor of 11 collective books, and author or co-author of 140 scientific articles.

Inhalt

Preface.- Preliminaries.- First Passage Across a Level.- Exit from an interval.- Expected local time.- Regulated Processes.- Algorithms.- Conclusion.- References.- Index.