The aim of this book is to present a recently developed approach suitable for investigating a variety of qualitative aspects of order-preserving random dynamical systems and to give the background for further development of the theory. The main objects considered are equilibria and attractors. The effectiveness of this approach is demonstrated by analysing the long-time behaviour of some classes of random and stochastic ordinary differential equations which arise in many applications.

The following notes grew out oflectures held during the DMV-Seminar on Random Media in November 1999 at the Mathematics Research Institute of Oberwolfach, and in February-March 2000 at the Ecole Normale Superieure in Paris. In both places the atmosphere was very friendly and stimulating. The positive response of the audience was encouragement enough to write up these notes. I hope they will carryover the enjoyment of the live lectures. I whole heartedly wish to thank Profs. Matthias Kreck and Jean-Franc;ois Le Gall who were respon sible for these two very enjoyable visits, Laurent Miclo for his comments on an earlier version of these notes, and last but not least Erwin Bolthausen who was my accomplice during the DMV-Seminar. A Brief Introduction The main theme of this series of lectures are "Random motions in random me dia." The subject gathers a variety of probabilistic models often originated from physical sciences such as solid state physics, physical chemistry, oceanography, biophysics . . ., in which typically some diffusion mechanism takes place in an inho mogeneous medium. Randomness appears at two levels. It comes in the description of the motion of the particle diffusing in the medium, this is a rather traditional point of view for probability theory; but it also comes in the very description of the medium in which the diffusion takes place."

This book reviews recent theoretical, computational and experimental developments in mechanics of random and multiscale solid materials. The aim is to provide tools for better understanding and prediction of the effects of stochastic (non-periodic) microstructures on materials' mesoscopic and macroscopic properties. Particular topics involve a review of experimental techniques for the microstructure description, a survey of key methods of probability theory applied to the description and representation of microstructures by random modes, static and dynamic elasticity and non-linear problems in random media via variational principles, stochastic wave propagation, Monte Carlo simulation of random continuous and discrete media, fracture statistics models, and computational micromechanics.

'Et moi, ..~ si lavait su CO.llUlJalt en revc:nir, One acMcc matbcmatica bu JaIdcred the human rac:c. It bu put COIDIDOD _ beet je n'y serais point aBe.' Jules Verne wbac it bdoup, 0Jl !be~ IbcII _t to !be dusty cauialcr Iabc&d 'diMardod__ The series is divergent; thc:reforc we may be -'. I!.ticT. Bc:I1 able to do something with it. O. Hcavisidc Mathematics is a tool for thought. A highly necessary tool in a world when: both feedback and non- linearities abound. Similarly. all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statcmalts as: 'One service topology has rendered mathematical physics ...*; 'One service logic has rendered c0m- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series. This series, Mathematics and Its Applications. started in 19n. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote "Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However. the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branc:hes. It also happens, quite often in fact, that branches which were thought to be completely.

This book constitutes the refereed proceedings of the International Symposium on Stochastic Algorithms: Foundations and Applications, SAGA 2001, held in Berlin, Germany in December 2001. The nine revised full papers presented together with four invited papers were carefully reviewed and selected for inclusion in the book. The papers are devoted to the design and analysis, experimental evaluation, and real-world application of stochasitc algorithms; in particular, new algorithmic ideas involving stochastic decisions and exploiting probabilistic properties of the underlying problem are introduced. Among the application fields are network and distributed algorithms, local search methods, and computational learning.

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.

In this book, Professor Pinsky gives a self-contained account of the theory of positive harmonic functions for second order elliptic operators, using an integrated probabilistic and analytic approach. The book begins with a treatment of the construction and basic properties of diffusion processes. This theory then serves as a vehicle for studying positive harmonic funtions. Starting with a rigorous treatment of the spectral theory of elliptic operators with nice coefficients on smooth, bounded domains, the author then develops the theory of the generalized principal eigenvalue, and the related criticality theory for elliptic operators on arbitrary domains. Martin boundary theory is considered, and the Martin boundary is explicitly calculated for several classes of operators. The book provides an array of criteria for determining whether a diffusion process is transient or recurrent. Also introduced are the theory of bounded harmonic functions, and Brownian motion on manifolds of negative curvature. Many results that form the folklore of the subject are here given a rigorous exposition, making this book a useful reference for the specialist, and an excellent guide for the graduate student.

The lift zonoid approach is based on a new representation of probability measures: a d-variate probability measure is represented by a convex set, its lift zonoid. First, lift zonoids are useful in data analysis to describe an empiricaldistribution by central (so-called trimmed) regions. They give rise to a concept of data depth related to the mean which is also useful in nonparametric tests for location and scale. Second, for comparing random vectors, the set inclusion of lift zonoids defines a stochastic order that reflects the dispersion of random vectors. This has many applications to stochastic comparison problems in economics and other fields. This monograph ves the first account in book form of the theory of lift zonoids and demonstrates its usefulness in multivariate analysis. Chapter 1 offers the reader an informal introduction to basic ideas, Chapter 2 presents a comprehensive investigation into the theory. The remaining seven chapters treat various applications of the lift zonoid approach and may be separately studied. Readers are assumed to have a firm grounding in probability at the graduate level. Karl Mosler is Professor of Statistics and Econometrics at the University of Cologne. He is Editor of the Allgemeines Statistisches Archive, Journal of the German Statistical Society, and has authored numerous research articles and four books (all with Springer-Verlag) in statistics and operations research.

The book provides an easily accessible computationally oriented introduction into the numerical solution of stochastic differential equations using computer experiments. It develops in the reader an ability to apply numerical methods solving stochastic differential equations in their own fields. Furthermore, it creates an intuitive understanding of the necessary theoretical background from stochastic and numeric analysis. The book is related to the more theoretical monograph P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, 1992, but can be independently used. It provides solutions to over 100 exercises used in this monograph to illustrate the theory. Corresponding Turbo Pascal programs are given on a floppy disk; furthermore commentaries on the programs and their use are carefully worked out in the book.

This graduate-level textbook covers modelling, programming and analysis of stochastic computer simulation experiments, including the mathematical and statistical foundations of simulation and why it works. The book is rigorous and complete, but concise and accessible, providing all necessary background material. Object-oriented programming of simulations is illustrated in Python, while the majority of the book is programming language independent. In addition to covering the foundations of simulation and simulation programming for applications, the text prepares readers to use simulation in their research. A solutions manual for end-of-chapter exercises is available for instructors.

In 1958, Stanford University Press published "Studies in the Mathematical Theory of Inventory and Production" (edited by Kenneth J. Arrow, Samuel Karlin, and Herbert Scarf), which became the pioneering road map for the next forty years of research in this area. One of the outgrowths of this research was development of the field of supply-chain management, which deals with the ways organizations can achieve competitive advantage by coordinating the activities involved in creating products--including designing, procuring, transforming, moving, storing, selling, providing after-sales service, and recycling. Following in this tradition, "Foundations of Stochastic Inventory Theory" has a dual purpose, serving as an advanced textbook designed to prepare doctoral students to do research on the mathematical foundations of inventory theory and as a reference work for those already engaged in such research.
The author begins by presenting two basic inventory models: the economic order quantity model, which deals with "cycle stocks," and the newsvendor model, which deals with "safety stocks." He then describes foundational concepts, methods, and tools that prepare the reader to analyze inventory problems in which uncertainty plays a key role. Dynamic optimization is an important part of this preparation, which emphasizes insights gained from studying the role of uncertainty, rather than focusing on the derivation of numerical solutions and algorithms (with the exception of two chapters on computational issues in infinite-horizon models).
All fourteen chapters in the book, and four of the five appendixes, conclude with exercises that either solidify or extend the concepts introduced. Some of these exercises have served as Ph.D. qualifying examination questions in the Operations, Information, and Technology area of the Stanford Graduate School of Business.

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 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 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."

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.

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.

This book presents stochastic processes in a comprehensive, user-friendly and accessible way containing numerous worked examples. The large number of exercises allows readers to check their understanding of the underlying theory, along with their ability to apply stochastic modelling in their own fields, making the book an excellent basis for self-study. It assumes basic knowledge of calculus and probability theory. The authors also include important proofs and theoretically challenging examples and exercises, thus making the book attractive to those whose interest is more mathematical.

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.

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.

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