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 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 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.

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/.

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.

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.

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."

This book presents a review of recent developments in the theory and construction of index numbers using the stochastic approach, demonstrating the versatility of this approach in handling various index number problems within a single conceptual framework. It also contains a brief, but complete, review of the existing approaches to index numbers with illustrative numerical examples.;The stochastic approach considers the index number problem as a signal extraction problem. The strength and reliability of the signal extracted from price and quantity changes for different commodities depends on the messages received and the information content of the messages. The most important applications of the new approach are to be found in the context of measuring rate of inflation and fixed and chain base index numbers for temporal comparisons and for spatial inter-country comparisons - the latter generally require special index number formulae that result in transitive and base invariant comparisons.

The chapters of this book were written by structural engineers. The approach, therefore, is not aiming toward a scientific modelling of the response but to the definition of engineering procedures for detecting and avoiding undesired phenomena. In this sense chaotic and stochastic behaviour can be tackled in a similar manner. This aspect is illustrated in Chapter 1. Chapters 2 and 3 are entirely devoted to Stochastic Dynamics and cover single-degree-of-freedom systems and impact problems, respectively. Chapter 4 provides details on the numerical tools necessary for evaluating the main indexes useful for the classification of the motion and for estimating the response probability density function. Chapter 5 gives an overview of random vibration methods for linear and nonlinear multi-degree-of-freedom systems. The randomness of the material characteristics and the relevant stochastic models ar considered in Chapter 6. Chapter 7, eventually, deals with large engineering sytems under stochastic excitation and allows for the stochastic nature of the mechanical and geometrical properties.

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.

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."

The 2-volume-book is an updated, reorganized and considerably enlarged version of the previous edition of the Research Problem Book in Analysis (LNM 1043), a collection familiar to many analysts, that has sparked off much research. This new edition, created in a joint effort by a large team of analysts, is, like its predecessor, a collection of unsolved problems of modern analysis designed as informally written mini-articles, each containing not only a statement of a problem but also historical and metho- dological comments, motivation, conjectures and discussion of possible connections, of plausible approaches as well as a list of references. There are now 342 of these mini- articles, almost twice as many as in the previous edition, despite the fact that a good deal of them have been solved!

The classical theory of natural selection, as developed by Fisher, Haldane, and 'Wright, and their followers, is in a sense a statistical theory. By and large the classical theory assumes that the underlying environment in which evolution transpires is both constant and stable - the theory is in this sense deterministic. In reality, on the other hand, nature is almost always changing and unstable. We do not yet possess a complete theory of natural selection in stochastic environ ments. Perhaps it has been thought that such a theory is unimportant, or that it would be too difficult. Our own view is that the time is now ripe for the development of a probabilistic theory of natural selection. The present volume is an attempt to provide an elementary introduction to this probabilistic theory. Each author was asked to con tribute a simple, basic introduction to his or her specialty, including lively discussions and speculation. We hope that the book contributes further to the understanding of the roles of "Chance and Necessity" (Monod 1971) as integrated components of adaptation in nature."

Stochastic processes with independent increments on a group are generalized to the concept of "white noise" on a Hopf algebra or bialgebra. The main purpose of the book is the characterization of these processes as solutions of quantum stochastic differential equations in the sense of R.L. Hudsonand K.R. Parthasarathy. The notes are a contribution to quantum probability but they are also related to classical probability, quantum groups, and operator algebras. The Az ma martingales appear as examples of white noise on a Hopf algebra which is a deformation of the Heisenberg group. The book will be of interest to probabilists and quantum probabilists. Specialists in algebraic structures who are curious about the role of their concepts in probablility theory as well as quantum theory may find the book interesting. The reader should havesome knowledge of functional analysis, operator algebras, and probability theory.

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.

Fatigue of engineering materials is a very complicated process that is difficult to accurately describe and predict. It is no doubt nowadays, that a fatigue of real materials should be regarded as a random phenomenon and analyzed by use of stochastic theory. This volume of the lectures sumarises the latest achievements in stochastic modelling and analysis of fatigue. The lectures cover the following important aspects of modern analysis of fatigue: methodology of stochastic modelling of fatigue, tools for characterization of random fatigue loads, physical and mechanical aspects of random fatigue, basic stochastic models for fatigue and the estimation of fatigue reliability of specific structural systems.

Stochastic Programming offers models and methods for decision problems wheresome of the data are uncertain. These models have features and structural properties which are preferably exploited by SP methods within the solution process. This work contributes to the methodology for two-stagemodels. In these models the objective function is given as an integral, whose integrand depends on a random vector, on its probability measure and on a decision. The main results of this work have been derived with the intention to ease these difficulties: After investigating duality relations for convex optimization problems with supply/demand and prices being treated as parameters, a stability criterion is stated and proves subdifferentiability of the value function. This criterion is employed for proving the existence of bilinear functions, which minorize/majorize the integrand. Additionally, these minorants/majorants support the integrand on generalized barycenters of simplicial faces of specially shaped polytopes and amount to an approach which is denoted barycentric approximation scheme.

All the papers contained in the volume are original, fully refereed researchpapers. They represent a fairly broad spectrum of the research activity in probability theory, which was done internationally in 1990-1991, with particular emphasis on Markov processes and stochastic calculus. The latter subject keeps growing, and some important new developments, included in the volume, concern anticipative stochastic integrals, and new applications of the enlargements of filtrations to the study of zeros of martingales. FROM THE CONTENTS: R. Bass, D. Khoshnevisan: Stochastic calculus and the continuity of local times of Levy processes.- M.T. Barlow, P. Imkeller: On some sample path properties of Skorokhod integral processes.- T.S. Mountford: A critical function for the planar Brownian convex hull.- L. Dubins, M. Smorodinsky: The modified, discrete Levy transformation is Bernoulli.- M. Baxter: Markov processes on the boundary of the binary tree.- R. Abraham: Unarbre aleatoire infini associe a l'excursion brownienne.- S.E. Kuznetsov: On the existence of a dual semigroup.