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.

The 31 papers collected here present original research results obtained in 1995-96, on Brownian motion and, more generally, diffusion processes, martingales, Wiener spaces, polymer measures.

In recent years, the classical theory of stochastic integration and stochastic differential equations has been extended to a non-commutative set-up to develop models for quantum noises. The author, a specialist of classical stochastic calculus and martingale theory, tries to provide an introduction to this rapidly expanding field in a way which should be accessible to probabilists familiar with the Ito integral. It can also, on the other hand, provide a means of access to the methods of stochastic calculus for physicists familiar with Fock space analysis. For this second edition, the author has added about 30 pages of new material, mostly on quantum stochastic integrals.

This book presents an algebraic development of the theory of countable state space Markov chains with discrete and continuous time parameters.

This book studies ergodic-theoretic aspects of random dynam- ical systems, i.e. of deterministic systems with noise. It aims to present a systematic treatment of a series of recent results concerning invariant measures, entropy and Lyapunov exponents of such systems, and can be viewed as an update of Kifer's book. An entropy formula of Pesin's type occupies the central part. The introduction of relation numbers (ch.2) is original and most methods involved in the book are canonical in dynamical systems or measure theory. The book is intended for people interested in noise-perturbed dynam- ical systems, and can pave the way to further study of the subject. Reasonable knowledge of differential geometry, measure theory, ergodic theory, dynamical systems and preferably random processes is assumed.

Evolution and learning in games is a topic of current intense interest. Evolution theory is widely viewed as one of the most promising approaches to understanding learning, bounded rationality, and change in complex social environments. This graduate textbook covers the recent developments with an emphasis on economic contexts and applications. Covering both deterministic and stochastic evolutionary dynamics which play an important role in evolutionary processes, it also includes the recent stochastic evolutionary framework that has been developed (and applied widely) in the last few years. The recent boom experienced by this discipline makes this book's systematic presentation of its essential contributions, using mathematical knowledge only when required, especially useful for any newcomer to the field. Packed with numerous economic applications of the theory, with suggestions for new avenues of research, it will prove invaluable to postgraduate economists.

Probabilistic methods can be applied very successfully to a number of asymptotic problems for second-order linear and non-linear partial differential equations. Due to the close connection between the second order differential operators with a non-negative characteristic form on the one hand and Markov processes on the other, many problems in PDE's can be reformulated as problems for corresponding stochastic processes and vice versa. In the present book four classes of problems are considered: - the Dirichlet problem with a small parameter in higher derivatives for differential equations and systems - the averaging principle for stochastic processes and PDE's - homogenization in PDE's and in stochastic processes - wave front propagation for semilinear differential equations and systems. From the probabilistic point of view, the first two topics concern random perturbations of dynamical systems. The third topic, homog- enization, is a natural problem for stochastic processes as well as for PDE's. Wave fronts in semilinear PDE's are interesting examples of pattern formation in reaction-diffusion equations. The text presents new results in probability theory and their applica- tion to the above problems. Various examples help the reader to understand the effects. Prerequisites are knowledge in probability theory and in partial differential equations.

This book provides a self-contained account of periodic models for seasonally observed economic time series with stochastic trends. Two key concepts are periodic integration and periodic cointegration. Periodic integration implies that a seasonally varying differencing filter is required to remove a stochastic trend. Periodic cointegration amounts to allowing cointegration paort-term adjustment parameters to vary with the season. The emphasis is on useful econrameters and shometric models that explicitly describe seasonal variation and can reasonably be interpreted in terms of economic behaviour. The analysis considers econometric theory, Monte Carlo simulation, and forecasting, and it is illustrated with numerous empirical time series. A key feature of the proposed models is that changing seasonal fluctuations depend on the trend and business cycle fluctuations. In the case of such dependence, it is shown that seasonal adjustment leads to inappropriate results.
About the Series
Advanced Texts in Econometrics is a distinguished and rapidly expanding series in which leading econometricians assess recent developments in such areas as stochastic probability, panel and time series data analysis, modeling, and cointegration. In both hardback and affordable paperback, each volume explains the nature and applicability of a topic in greater depth than possible in introductory textbooks or single journal articles. Each definitive work is formatted to be as accessible and convenient for those who are not familiar with the detailed primary literature.

In order to obtain more reliable optimal solutions of concrete technical/economic problems, e.g. optimal design problems, the often known stochastic variations of many technical/economic parameters have to be taken into account already in the planning phase. Hence, ordinary mathematical programs have to be replaced by appropriate stochastic programs. New theoretical insight into several branches of reliability-oriented optimization of stochastic systems, new computational approaches and technical/economic applications of stochastic programming methods can be found in this volume.

This book is an introductionary course in stochastic ordering and dependence in the field of applied probability for readers with some background in mathematics. It is based on lectures and senlinars I have been giving for students at Mathematical Institute of Wroclaw University, and on a graduate course a.t Industrial Engineering Department of Texas A&M University, College Station, and addressed to a reader willing to use for example Lebesgue measure, conditional expectations with respect to sigma fields, martingales, or compensators as a common language in this field. In Chapter 1 a selection of one dimensional orderings is presented together with applications in the theory of queues, some parts of this selection are based on the recent literature (not older than five years). In Chapter 2 the material is centered around the strong stochastic ordering in many dimen sional spaces and functional spaces. Necessary facts about conditioning, Markov processes an"d point processes are introduced together with some classical results such as the product formula and Poissonian departure theorem for Jackson networks, or monotonicity results for some re newal processes, then results on stochastic ordering of networks, re ment policies and single server queues connected with Markov renewal processes are given. Chapter 3 is devoted to dependence and relations between dependence and ordering, exem plified by results on queueing networks and point processes among others."

The aim of this monograph is to show how random sums (that is, the summation of a random number of dependent random variables) may be used to analyse the behaviour of branching stochastic processes. The author shows how these techniques may yield insight and new results when applied to a wide range of branching processes. In particular, processes with reproduction-dependent and non-stationary immigration may be analysed quite simply from this perspective. On the other hand some new characterizations of the branching process without immigration dealing with its genealogical tree can be studied. Readers are assumed to have a firm grounding in probability and stochastic processes, but otherwise this account is self-contained. As a result, researchers and graduate students tackling problems in this area will find this makes a useful contribution to their work.

This book consists of two strongly interweaved parts: the mathematical theory of stochastic processes and its applications to molecular theories of polymeric fluids. The comprehensive mathematical background provided in the first section will be equally useful in many other branches of engineering and the natural sciences. The second part provides readers with a more direct understanding of polymer dynamics, allowing them to identify exactly solvable models more easily, and to develop efficient computer simulation algorithms in a straightforward manner. In view of the examples and applications to problems taken from the front line of science, this volume may be used both as a basic textbook or as a reference book. Program examples written in FORTRAN are available via ftp from ftp.springer.de/pub/chemistry/polysim/.

The present monograph is a comprehensive summary of the research on visibility in random fields, which I have conducted with the late Professor Micha Yadin for over ten years. This research, which resulted in several published papers and technical reports (see bibliography), was motivated by some military problems, which were brought to our attention by Mr. Pete Shugart of the US Army TRADOC Systems Analysis Activity, presently called US Army TRADOC Analysis Command. The Director ofTRASANA at the time, the late Dr. Wilbur Payne, identified the problems and encouraged the support and funding of this research by the US Army. Research contracts were first administered through the Office of Naval Research, and subsequently by the Army Research Office. We are most grateful to all involved for this support and encouragement. In 1986 I administered a three-day workshop on problem solving in the area of sto chastic visibility. This workshop was held at the White Sands Missile Range facility. A set of notes with some software were written for this workshop. This workshop led to the incorporation of some of the methods discussed in the present book into the Army simulation package CASTFOREM. Several people encouraged me to extend those notes and write the present monograph on the level of those notes, so that the material will be more widely available for applications."

Since its first publication in 1965 in the series "Grundlehren der mathematischen Wissenschaften" this book has had a profound and enduring influence on research into the stochastic processes associated with diffusion phenomena. Generations of mathematicians have appreciated the clarity of the descriptions given of one- or more- dimensional diffusion processes and the mathematical insight provided into Brownian motion. Now, with its republication in the "Classics in Mathematics" it is hoped that a new generation will be able to enjoy the classic text of Ito and McKean."""

In this volume of original research papers, the main topics discussed relate to the asymptotic windings of planar Brownian motion, structure equations, closure properties of stochastic integrals. The contents of the volume represent an important fraction of research undertaken by French probabilists and their collaborators from abroad during the academic year 1992-1993.

Probabilistic models of technical systems are studied here whose finite state space is partitioned into two or more subsets. The systems considered are such that each of those subsets of the state space will correspond to a certain performance level of the system. The crudest approach differentiates between 'working' and 'failed' system states only. Another, more sophisticated, approach will differentiate between the various levels of redundancy provided by the system. The dependability characteristics examined here are random variables associated with the state space's partitioned structure; some typical ones are as follows * The sequence of the lengths of the system's working periods; * The sequences of the times spent by the system at the various performance levels; * The cumulative time spent by the system in the set of working states during the first m working periods; * The total cumulative 'up' time of the system until final breakdown; * The number of repair events during a fmite time interval; * The number of repair events until final system breakdown; * Any combination of the above. These dependability characteristics will be discussed within the Markov and semi-Markov frameworks.

In three chapters on Exponential Martingales, BMO-martingales, and Exponential of BMO, this book explains in detail the beautiful properties of continuous exponential martingales that play an essential role in various questions concerning the absolute continuity of probability laws of stochastic processes. The second and principal aim is to provide a full report on the exciting results on BMO in the theory of exponential martingales. The reader is assumed to be familiar with the general theory of continuous martingales.

White Noise Calculus is a distribution theory on Gaussian space, proposed by T. Hida in 1975. This approach enables us to use pointwise defined creation and annihilation operators as well as the well-established theory of nuclear space.This self-contained monograph presents, for the first time, a systematic introduction to operator theory on fock space by means of white noise calculus. The goal is a comprehensive account of general expansion theory of Fock space operators and its applications. In particular, first order differential operators, Laplacians, rotation group, Fourier transform and their interrelations are discussed in detail w.r.t. harmonic analysis on Gaussian space. The mathematical formalism used here is based on distribution theory and functional analysis, prior knowledge of white noise calculus is not required.