This book shows how techniques from the perturbation theory of operators, applied to a quasi-compact positive kernel, may be used to obtain limit theorems for Markov chains or to describe stochastic properties of dynamical systems.A general framework for this method is given and then applied to treat several specific cases. An essential element of this work is the description of the peripheral spectra of a quasi-compact Markov kernel and of its Fourier-Laplace perturbations. This is first done in the ergodic but non-mixing case. This work is extended by the second author to the non-ergodic case.The only prerequisites for this book are a knowledge of the basic techniques of probability theory and of notions of elementary functional analysis.

Discrete-time Stochastic Systems gives a comprehensive introduction to the estimation and control of dynamic stochastic systems and provides complete derivations of key results such as the basic relations for Wiener filtering. The book covers both state-space methods and those based on the polynomial approach. Similarities and differences between these approaches are highlighted. Some non-linear aspects of stochastic systems (such as the bispectrum and extended Kalman filter) are also introduced and analysed. The books chief features are as follows: inclusion of the polynomial approach provides alternative and simpler computational methods than simple reliance on state-space methods; algorithms for analysis and design of stochastic systems allow for ease of implementation and experimentation by the reader; the highlighting of spectral factorization gives appropriate emphasis to this key concept often overlooked in the literature; explicit solutions of Wiener problems are handy schemes, well suited for computations compared with more commonly available but abstract formulations; complex-valued models that are directly applicable to many problems in signal processing and communications. Changes in the second edition include: additional information covering spectral factorisation and the innovations form; the chapter on optimal estimation being completely rewritten to focus on a posterior estimates rather than maximum likelihood; new material on fixed lag smoothing and algorithms for solving Riccati equations are improved and more up to date; new presentation of polynomial control and new derivation of linear-quadratic-Gaussian control. Discrete-time Stochastic Systems is primarily of benefit to students taking M.Sc. courses in stochastic estimation and control, electronic engineering and signal processing but may also be of assistance for self study and as a reference.

During the last two decades, in several branches of science (water waves, crystal growth, travelling waves in one dimensional lattices, splitting of separatrices, ...) different problems appeared in which the key point is the computation of exponentially small terms. This self-contained monograph gives new and rigorous mathematical tools which enable a systematic study of such problems. Starting with elementary illuminating examples, the book contains (i) new asymptotical tools for obtaining exponentially small equivalents of oscillatory integrals involving solutions of nonlinear differential equations; (ii) implementation of these tools for solving old open problems of bifurcation theory such as existence of homoclinic connections near resonances in reversible systems

This book describes stochastic epidemic models and methods for statistically analyzing them. It is aimed at statisticians, biostatisticians, and biomathematicians.

During the last decade, problems in the world of finance have been the main driving force for developing sophisticated mathematical methods which may be used for identifying and measuring risk. The focus is still on quantifying market and credit risk, but general operational risks will become more important in the future. In this book the reader will find approaches from economic theory, allocation problems, credit scoring, volatility structures, general market risk, country risk and extreme value theory. The contributions of this book reflect the views of leading practitioners and academics in the field of risk management.
Most of the models considered for the evolution of asset values are of a complex and stochastic nature, including stochastic volatility models in continuous time as well as their counterparts in discrete time, the family of GARCH-like time series. The contents reflect the fact that a major part of recent research has been motivated by applications in finance, but most of the mathematical approaches may be used for risk analysis in engineering and science in a rather straightforward manner. As known from insurance mathematics for some time, extreme damages from natural disaster follow similar stochastic laws as extreme losses from certain investments.
The articles discuss critical concepts such as value-at-risk, volatility and other risk masures in nonstandard situations. Stochastic processes beyond geometric Brownian motion allow for a more realistic reflection of stylized facts like leptokurtosis or skewness of return distrubutions which often are observed in real data. Procedures for detecting change points in time series allow for dealing with the risk of a sudden structural change of the market. Models for extremal events in financial time series or stochastic processes in continuous time are of prime importance for risk management as, in practice, these rare events frequently dominate the whole profit/loss-process.

The average-case analysis of numerical problems is the counterpart of the more traditional worst-case approach. The analysis of average error and cost leads to new insight on numerical problems as well as to new algorithms. The book provides a survey of results that were mainly obtained during the last 10 years and also contains new results. The problems under consideration include approximation/optimal recovery and numerical integration of univariate and multivariate functions as well as zero-finding and global optimization. Background material, e.g. on reproducing kernel Hilbert spaces and random fields, is provided.

The monograph is devoted mainly to the analytical study of the differential, pseudo-differential and stochastic evolution equations describing the transition probabilities of various Markov processes. These include (i) diffusions (in particular,degenerate diffusions), (ii) more general jump-diffusions, especially stable jump-diffusions driven by stable Lévy processes, (iii) complex stochastic Schrödinger equations which correspond to models of quantum open systems. The main results of the book concern the existence, two-sided estimates, path integral representation, and small time and semiclassical asymptotics for the Green functions (or fundamental solutions) of these equations, which represent the transition probability densities of the corresponding random process. The boundary value problem for Hamiltonian systems and some spectral asymptotics ar also discussed. Readers should have an elementary knowledge of probability, complex and functional analysis, and calculus.

The book offers an accessible reference for researchers in the probability, statistics and special functions communities. It gives a variety of interdisciplinary relatiions between the two main ingredients of stochastic processes and orthogonal polynomials. It covers topics like time dependent and asymptotic anlaysis for birth-death processes and diffusions, martingale relations for Lévy processes, stochastic integrals and Stein's approximation method. Almost all well-known orthogonal polynomials, which are brought together in the so-called Askey Scheme, come into play.
This volume clearly illustrates the powerful mathematical role of orthogonal polynomials in the analysis of stochastic processes and is made accessible for all mathematicians with a basic background in probability theory and mathematical analysis.
Wim Schoutens is a Postdoctoral Researcher of the Fund for Scientific Research-Flanders (Belgium). He received his PhD in Science from the Catholic University of Leuven, Belgium.

The book focuses on stochastic modeling of population processes. The book presents new symbolic mathematical software to develop practical methodological tools for stochastic population modeling. The book assumes calculus and some knowledge of mathematical modeling, including the use of differential equations and matrix algebra.

This book studies the foundations of the theory of linear and nonlinear forms in single and multiple random variables including the single and multiple random series and stochastic integrals, both Gaussian and non-Gaussian. This subject is intimately connected with a number of classical problems of probability theory such as the summation of independent random variables, martingale theory, and Wiener's theory of polynomial chaos. The book contains a number of older results as well as more recent, or previously unpublished, results. The emphasis is on domination principles for comparison of different sequences of random variables and on decoupling techniques. These tools prove very useful in many areas ofprobability and analysis, and the book contains only their selected applications. On the other hand, the use of the Fourier transform - another classical, but limiting, tool in probability theory - has been practically eliminated. The book is addressed to researchers and graduate students in prob ability theory, stochastic processes and theoretical statistics, as well as in several areas oftheoretical physics and engineering. Although the ex position is conducted - as much as is possible - for random variables with values in general Banach spaces, we strive to avoid methods that would depend on the intricate geometric properties of normed spaces. As a result, it is possible to read the book in its entirety assuming that all the Banach spaces are simply finite dimensional Euclidean spaces."

Stochastic differential equations, and Hoermander form representations of diffusion operators, can determine a linear connection associated to the underlying (sub)-Riemannian structure. This is systematically described, together with its invariants, and then exploited to discuss qualitative properties of stochastic flows, and analysis on path spaces of compact manifolds with diffusion measures. This should be useful to stochastic analysts, especially those with interests in stochastic flows, infinite dimensional analysis, or geometric analysis, and also to researchers in sub-Riemannian geometry. A basic background in differential geometry is assumed, but the construction of the connections is very direct and itself gives an intuitive and concrete introduction. Knowledge of stochastic analysis is also assumed for later chapters.

This book describes the representations of Lie superalgebras that are yielded by a graded version of Hudson-Parthasarathy quantum stochastic calculus. Quantum stochastic calculus and grading theory are given concise introductions, extending readership to mathematicians and physicists with a basic knowledge of algebra and infinite-dimensional Hilbert spaces. The develpment of an explicit formula for the chaotic expansion of a polynomial of quantum stochastic integrals is particularly interesting. The book aims to provide a self-contained exposition of what is known about Z_2-graded quantum stochastic calculus and to provide a framework for future research into this new and fertile area.

The book discusses the estimation theory for the wide class of inhomogeneous Poisson processes. The consistency, limit distributions and the convergence of moments of parameter estimators are established in regular and non-regular (change-point type) problems. The maximum likelihood, Bayesian, and the minimum distance estimators are investigated in parametric problems and the empiric intensity measure and the kernel-type estimators are studied in nonparametric estimation problems. The properties of the estimators are also described in the situations when the observed Poisson process does not belong to the parametric family (no true model), when there are many true models (nonidentifiable family), when the observation window can be chosen by an optimal way, and others. The question of asymptotic efficiency of estimators is discussed in all of these problems. The book will be useful for those who use models of Poisson processes in their research. The large number of examples of inhomogeneous Poisson processes discussed in the book are taken from the fields of optical communications, reliability, image processing, and nuclear medicine. The material is suitable for graduate courses on stochastic processes. The book assumes familiarity with probability theory and mathematical statistics. Yury A. Kutoyants, Professor of Mathematics at the University of Main, Le Mans, France, is a member of the Bernoulli Society, the Mathematical Society of France, and the Institute of Mathematical Statistics. He is associate editor of "Finance and Stochastics" and "Statistical Inference for Stochastic Processes." He is author of "Parameter Estimation for Stochastic Processes" (Heldermann Verlag, Berlin, 1984)and "Identification of Dynamical Systems with Small Noise" (Kluwer, Dordrecht, 1994), and the of about 70 articles on the

2020 Taylor & Francis Award Winner for Outstanding New Textbook! Featuring recent advances in the field, this new textbook presents probability and statistics, and their applications in stochastic processes. This book presents key information for understanding the essential aspects of basic probability theory and concepts of reliability as an application. The purpose of this book is to provide an option in this field that combines these areas in one book, balances both theory and practical applications, and also keeps the practitioners in mind. Features Includes numerous examples using current technologies with applications in various fields of study Offers many practical applications of probability in queueing models, all of which are related to the appropriate stochastic processes (continuous time such as waiting time, and fuzzy and discrete time like the classic Gambler's Ruin Problem) Presents different current topics like probability distributions used in real-world applications of statistics such as climate control and pollution Different types of computer software such as MATLAB (R), Minitab, MS Excel, and R as options for illustration, programing and calculation purposes and data analysis Covers reliability and its application in network queues

This book is devoted to the theory and applications of nonparametic functional estimation and prediction. Chapter 1 provides an overview of inequalities and limit theorems for strong mixing processes. Density and regression estimation in discrete time are studied in Chapter 2 and 3. The special rates of convergence which appear in continuous time are presented in Chapters 4 and 5. This second edition is extensively revised and it contains two new chapters. Chapter 6 discusses the surprising local time density estimator. Chapter 7 gives a detailed account of implementation of nonparametric method and practical examples in economics, finance and physics. Comarison with ARMA and ARCH methods shows the efficiency of nonparametric forecasting. The prerequisite is a knowledge of classical probability theory and statistics. Denis Bosq is Professor of Statistics at the Unviersity of Paris 6 (Pierre et Marie Curie). He is Editor-in-Chief of "Statistical Inference for Stochastic Processes" and an editor of "Journal of Nonparametric Statistics". He is an elected member of the International Statistical Institute. He has published about 90 papers or works in nonparametric statistics and four books.

This monograph proposes several approaches to convergence monitoring for MCMC algorithms which are centered on the theme of discrete Markov chains. After a short introduction to MCMC methods, including recent developments like perfect simulation and Langevin Metropolis-Hastings algorithms, and to the current convergence diagnostics, the contributors present the theoretical basis for a study of MCMC convergence using discrete Markov chains and their specificities. The contributors stress in particular that this study applies in a wide generality, starting with latent variable models like mixtures, then extending the scope to chains with renewal properties, and concluding with a general Markov chain. They then relate the different connections with discrete or finite Markov chains with practical convergence diagnostics which are either graphical plots (allocation map, divergence graph, variance stabilizing, normality plot), stopping rules (normality, stationarity, stability tests), or confidence bounds (divergence, asymptotic variance, normality). Most of the quantitative tools take advantage of manageable versions of the CLT. The different methods proposed here are first evaluated on a set of benchmark examples and then studied on three full scale realistic applications, along with the standard convergence diagnostics: A hidden Markov modelling of DNA sequences, including a perfect simulation implementation, a latent stage modelling of the dynamics of HIV infection, and a modelling of hospitalization duration by exponential mixtures. The monograph is the outcome of a monthly research seminar held at CREST, Paris, since 1995. The seminar involved the contributors to this monograph and wasled by Christian P. Robert, Head of the Satistics Laboratory at CREST and Professor of Statistics at the University of Rouen since 1992.

This volume contains the papers presented at the3rd International Wo- shoponRandomizationandApproximationTechniquesinComputer Science (RANDOM 99) and the 2nd International Workshop on - proximation Algorithms for Combinatorial Optimization Problems (APPROX 99), which took place concurrently at the University of California, Berkeley, from August 8 11, 1999. RANDOM 99 is concerned with appli- tions of randomness to computational and combinatorial problems, and is the third workshop in the series following Bologna (1997) and Barcelona (1998). APPROX 99 focuses on algorithmic and complexity issues surrounding the - velopment of e?cient approximate solutions to computationally hard problems, and is the second in the series after Aalborg (1998). The volume contains 24 contributed papers, selected by the two program committees from 44 submissions received in response to the call for papers, together with abstracts of invited lectures by Uri Feige (Weizmann Institute), Christos Papadimitriou (UC Berkeley), Madhu Sudan (MIT), and Avi Wigd- son (Hebrew University and IAS Princeton). We would like to thank all of the authors who submitted papers, our invited speakers, the external referees we consulted and the members of the program committees, who were: RANDOM 99 APPROX 99 Alistair Sinclair, UC Berkeley Dorit Hochbaum, UC Berkeley Noga Alon, Tel Aviv U. Sanjeev Arora, Princeton U. Jennifer Chayes, Microsoft Leslie Hall, Johns Hopkins U. Monika Henzinger, Compaq-SRC Samir Khuller, U. of Maryland Mark Jerrum, U. of Edinburgh Phil Klein, Brown U."

Senior probabilists from around the world with widely differing specialities gave their visions of the state of their specialty, why they think it is important, and how they think it will develop in the new millenium. The volume includes papers given at a symposium at Columbia University in 1995, but papers from others not at the meeting were added to broaden the coverage of areas. All papers were refereed.

Financial Mathematics is an exciting, emerging field of application. The five sets of course notes in this book provide a bird's eye view of the current "state of the art" and directions of research. For graduate students it will therefore serve as an introduction to the field while reseachers will find it a compact source of reference. The reader is expected to have a good knowledge of the basic mathematical tools corresponding to an introductory graduate level and sufficient familiarity with probabilistic methods, in particular stochastic analysis.

3 On the Economic Relevance of Rational Bubbles 79 3. 1 Capital markets . . . . . . . . . 80 3. 1. 1 Efficient capital markets 86 3. 1. 2 Rational bubbles on capital markets. 93 3. 1. 3 Economic caveats . 103 3. 2 Foreign exchange markets 109 3. 3 Hyperinflation. . . . . . . 117 4 On Testing for Rational Bubbles 123 4. 1 Indirect tests . . . . . . . . . 123 4. 1. 1 Variance bounds tests 124 4. 1. 2 Specification tests . . . 137 4. 1. 3 Integration and cointegration tests 140 4. 1. 4 Final assessment of indirect tests . 150 4. 1. 5 A digression: Charemza, Deadman (1995) analysis. 151 4. 2 Direct tests . . . . . . . . . . . . . . . . . . . . . . . . 157 4. 2. 1 Deterministic bubble in German hyperinflation. 158 4. 2. 2 Intrinsic bubbles on stock markets. 163 4. 2. 3 An econometric caveat . . . . . 168 4. 2. 4 Final assessment of direct tests 172 5 On the Explanatory Power of Rational Bubbles on the G- man Stock Market 175 5. 1 Data . . . . . . . 175 5. 2 Direct test for rational bubbles 181 5. 2. 1 Temporary Markovian bubbles. 184 5. 2. 2 Temporary intrinsic bubbles . . 193 ix 5. 2. 3 Permanent intrinsic bubbles 198 5. 3 A digression: Testing for unit roots 204 6 Concluding Remarks 215 A Results 221 A. 1 Temporary markovian bubbles. 221 A. 2 Temporary intrinsic bubbles . . 225 A. 3 Permanent intrinsic bubbles - Class 1 to 2 229 A. 4 Permanent intrinsic bubbles - Class 3 to 6 230 A. 5 Integration tests. . . . . . . . . . . . . . .

Two of the most exciting topics of current research in stochastic networks are the complementary subjects of stability and rare events - roughly, the former deals with the typical behavior of networks, and the latter with significant atypical behavior. Both are classical topics, of interest since the early days of queueing theory, that have experienced renewed interest mo tivated by new applications to emerging technologies. For example, new stability issues arise in the scheduling of multiple job classes in semiconduc tor manufacturing, the so-called "re-entrant lines;" and a prominent need for studying rare events is associated with the design of telecommunication systems using the new ATM (asynchronous transfer mode) technology so as to guarantee quality of service. The objective of this volume is hence to present a sample - by no means comprehensive - of recent research problems, methodologies, and results in these two exciting and burgeoning areas. The volume is organized in two parts, with the first part focusing on stability, and the second part on rare events. But it is impossible to draw sharp boundaries in a healthy field, and inevitably some articles touch on both issues and several develop links with other areas as well. Part I is concerned with the issue of stability in queueing networks."

In a competitive world, research in manufacturing systems plays an important role in creating, updating and improving the technologies and management practices of the economy. This volume presents some of the most recent results in stochastic manufacturing systems. Experts from the fields of applied mathematics, engineering and management sciences review and substantially update the recent advances in the control and optimization of manufacturing systems. Recent Advances in Control and Optimization of Manufacturing Systems consists of eight chapters divided into three parts which focus on Optimal Production Planning, Scheduling and Improvability and Approximate Optimality and Robustness. This book is intended for researchers and practitioners in the fields of systems theory, control and optimization, and operation management as well as in applied probability and stochastic processes.

This is author-approved bcc: This book provides a comprehensive treatment of linear mixed models, a technique devised to analyze continuous correlated data. It focuses on examples from designed experiments and longitudinal studies. The target audience includes applied statisticians and biomedical researchers in industry, public health organizations, contract research organizations, and academia. The book is explanatory rather than mathematical rigorous. Although most analyses were done with the MIXED procedure of the SAS software package, and many of its features are clearly elucidated, considerable effort was spent in presenting the data analyses in a software-independent fashion. Geert Verbeke is Assistant Professor at the Biostatistical Centre for Clinical Trials of the Katholieke Universiteit Leuven in Belgium. He received the B.S. degree in mathematics (1989) from the Katholieke Universiteit Leuven, the M.S. in biostatistics (1992) from the Limburgs Universitair Centrum, and earned a PhD in biostatistics (1995) from the Katholieke Universiteit Leuven. Dr. Verkeke wrote his dissertation, as well as a number of methodological articles, on various aspects on linear mixed models for longitudinal data analysis. He has held visiting positions at the Gerontology Researh Center and the Johns Hopkins University (Baltimore, MD). Geert Molenberghs is Assistant Professor of Biostatistics at the Limburgs Universitair Centrum in Belgium. He received the B.S. degree in mathematics (1988) and a PhD in biostatistics (1993) from the Universiteit Antwerpen. Dr. Molenberghs published methodological work on the analysis of non-response, and non-compliance in clinical trials. He serves as an associateeditor for Biometrics and Applied

This monograph contains some ofthe papers presented at a UK-Japanese Workshop on Stochastic Modelling in Innovative Manufacturing held at Churchill College, Cambridge on July 20 and 21st 1995, sponsored jointly by the UK Engineering and Physical Science Research Council and the British Council. Attending were 19 UK and 24 Japanese delegates representing 28 institutions. The aim of the workshop was to discuss the modelling work being done by researchers in both countries on the new activities and challenges occurring in manufacturing. These challenges have arisen because of the increasingly uncertain environment of modern manufacturing due to the commercial need to respond more quickly to customers demands, and the move to just-in-time manufacturing and flexible manufacturing systems and the increasing requirements for quality. As well as time pressure, the increasing importance of the quality of the products, the need to hold the minimum stock of components, and the importance of reliable production systems has meant that manufacturers need to design production systems that perform well in randomly varying conditions and that their operating procedures can respond to changes in conditions and requirements. This has increased the need to understand how manufacturing systems work in the random environments, and so emphasised the importance of stochastic models of such systems.

In the last decade there has been a steadily growing need for and interest in computational methods for solving stochastic optimization problems with or wihout constraints. Optimization techniques have been gaining greater acceptance in many industrial applications, and learning systems have made a significant impact on engineering problems in many areas, including modelling, control, optimization, pattern recognition, signal processing and diagnosis. Learning automata have an advantage over other methods in being applicable across a wide range of functions. Featuring new and efficient learning techniques for stochastic optimization, and with examples illustrating the practical application of these techniques, this volume will be of benefit to practicing control engineers and to graduate students taking courses in optimization, control theory or statistics.