Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes

ISBN-13: 9780817642150 / Angielski / Twarda / 2005 / 328 str.

Martin Jacobsen

Opis

The book aims at presenting a detailed and mathematically rigorous exposition of the theory and applications of a class of point processes and piecewise deterministic p- cesses. The framework is suf?ciently general to unify the treatment of several classes of stochastic phenomena: point processes, Markov chains and other Markov processes in continuous time, semi-Markov processes, queueing and storage models, and li- lihood processes. There are applications to ?nance, insurance and risk, population models, survival analysis, and congestion models. A major aim has been to show the versatility of piecewise deterministic Markov processes for applications and to show how they may also become useful in areas where thus far they have not been much in evidence. Originally the plan was to develop a graduate text on marked point processes - dexed by time which would focus on probabilistic structure and be essentially se- contained. However, it soon became apparent that the discussion should naturally include a traditional class of continuous time stochastic processes constructed from certain marked point processes. This class consists of 'piecewise deterministic p- cesses'; that is, processes with ?nitely many jumps on ?nite time intervals which, roughly speaking, develop deterministically between the random jump times. The - position starts with the point process theory and then uses this to treat the piecewise deterministic processes.

Recenzje

From the reviews:

"The most remarkable aspect of the book is the reader-friendly structure and the style in which it has been written. The importance of the book both from the practical and from the theoretical standpoint is unquestionable. This book will be an essential part of every mathematical library."   -ZENTRALBLATT Math

"This book consists of three parts: (I) Theory, (II) Applications and (III) Appendices, that concisely summarize necessary mathematical concepts which are often found scattered among numerous texts. ... The most remarkable aspect of the book is its reader-friendly structure and the style in which it has been written. The importance of this book, both from the practical and from the theoretical standpoint, is unquestionable. It will be an essential part of every mathematical library." (V. K. Oganyan, Mathematical Reviews, Issue 2007 a)

"This book deals with marked point processes on the positive real axis. It is rigorously written and largely self-contained. ... Each chapter starts with an appealing introduction, including hints to the most important references. The reader is guided by clearly stated comments and hints. ... Many examples and exercises are given ... . This book is suitable for advanced graduate and postgraduate students in mathematics with some preknowledge in this area. It will also be valuable for preparing a respective lecture course." (Martin Schlather, Journal of the American Statistical Association, Vol. 102 (479), 2007)

"The book will serve as a guide to the comprehensive study of the subject, and-more likely-as a reference book, inevitably a standard one. ... Martin Jacobsen write an accessible book is a true gift to the mathematics community. ... This is a good book, and an important one. ... Every researcher with even a passing interest in the subject will want this book on his or her shelf." (Philip Protter, SIAM Review, Vol. 49 (1), 2007)

Spis treści

* Preface Part I: Theory * Introduction * Simple and Marked Point Processes * Construction of SPPs and MPPs * Compensators and Mrtingales * Likelihood Processes * Independence * Piecewise Deterministic Markov Processes (PDMPs) Part II: Applications * Survival Analysis * Branching, Ruin, Soccer * A Model from Finance * Examples of Queueing Models Part III: Appendices * Appendix A: Differentiation of Cadlag Functions * Appendix B: Filtrations, Processes, Martingales * Bibliographical Notes * References * Notation Index * Index

Blurb

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.