Bøger i Probability and Its Applicatio serien

Overview This book is intended as a textbook in probability for graduate students in math­ ematics and related areas such as statistics, economics, physics, and operations research. Probability theory is a 'difficult' but productive marriage of mathemat­ ical abstraction and everyday intuition, and we have attempted to exhibit this fact. Thus we may appear at times to be obsessively careful in our presentation of the material, but our experience has shown that many students find them­ selves quite handicapped because they have never properly come to grips with the subtleties of the definitions and mathematical structures that form the foun­ dation of the field. Also, students may find many of the examples and problems to be computationally challenging, but it is our belief that one of the fascinat­ ing aspects of prob ability theory is its ability to say something concrete about the world around us, and we have done our best to coax the student into doing explicit calculations, often in the context of apparently elementary models. The practical applications of probability theory to various scientific fields are far-reaching, and a specialized treatment would be required to do justice to the interrelations between prob ability and any one of these areas. However, to give the reader a taste of the possibilities, we have included some examples, particularly from the field of statistics, such as order statistics, Dirichlet distri­ butions, and minimum variance unbiased estimation.

This text offers a mathematically rigorous exposition of the basic theory of marked point processes developing randomly over time, and shows how this theory may be used to treat piecewise deterministic stochastic processes in continuous time.The focus is on point processes that generate only finitely many points in finite time intervals, resulting in piecewise deterministic processes with "few jumps". The point processes are constructed from scratch with detailed proofs and their distributions characterized using compensating measures and martingale structures. Piecewise deterministic processes are defined and identified with certain marked point processes, which are then used in particular to construct and study a large class of piecewise deterministic Markov processes, whether time homogeneous or not.The second part of the book addresses applications of the just developed theory. This analysis of various models in applied statistics and probability includes examples and exercises in survival analysis, branching processes, ruin probabilities, sports (soccer), finance and risk management (arbitrage and portfolio trading strategies), and queueing theory.Graduate students and researchers interested in probabilistic modeling and its applications will find this text an excellent resource, requiring for mastery a solid foundation in probability theory, measure and integration, as well as some knowledge of stochastic processes and martingales. However, an explanatory introduction to each chapter highlights those portions that are crucial and those that can be omitted by non-specialists, making the material more accessible to a wider cross-disciplinary audience.