Biane, Philippe: Non-commutative stochastic calculus.-Voiculescu, Dan-Virgil: Lectures on free probability.- Guionnet, Alice: Large random matrices: Lectures on macroscopic asymptotics. "

Provides a Solid Foundation for Statistical Modeling and Inference and Demonstrates Its Breadth of Applicability Stochastic Modeling and Mathematical Statistics: A Text for Statisticians and Quantitative Scientists addresses core issues in post-calculus probability and statistics in a way that is useful for statistics and mathematics majors as well as students in the quantitative sciences. The book's conversational tone, which provides the mathematical justification behind widely used statistical methods in a reader-friendly manner, and the book's many examples, tutorials, exercises and problems for solution, together constitute an effective resource that students can read and learn from and instructors can count on as a worthy complement to their lectures. Using classroom-tested approaches that engage students in active learning, the text offers instructors the flexibility to control the mathematical level of their course. It contains the mathematical detail that is expected in a course for "majors" but is written in a way that emphasizes the intuitive content in statistical theory and the way theoretical results are used in practice. More than 1000 exercises and problems at varying levels of difficulty and with a broad range of topical focus give instructors many options in assigning homework and provide students with many problems on which to practice and from which to learn.

The book provides an introduction to advanced topics in stochastic processes and related stochastic analysis, and combines them with a sound presentation of the fundamentals of financial mathematics. It is wide-ranging in content, while at the same time placing much emphasis on good readability, motivation, and explanation of the issues covered.  Financial mathematical topics are first introduced in the context of discrete time processes and then transferred to continuous-time models. The basic construction of the stochastic integral and the associated martingale theory provide fundamental methods of the theory of stochastic processes for the construction of suitable stochastic models of financial mathematics, e.g. using stochastic differential equations. Central results of stochastic analysis such as the Itô formula, Girsanov's theorem and martingale representation theorems are of fundamental importance in financial mathematics, e.g. for the risk-neutral valuation formula (Black-Scholes formula) or the question of the hedgeability of options and the completeness of market models. Chapters on the valuation of options in complete and incomplete markets and on the determination of optimal hedging strategies conclude the range of topics. Advanced knowledge of probability theory is assumed, in particular of discrete-time processes (martingales, Markov chains) and continuous-time processes (Brownian motion, Lévy processes, processes with independent increments, Markov processes). The book is thus suitable for advanced students as a companion reading and for instructors as a basis for their own courses.This book is a translation of the original German 1st edition Stochastische Prozesse und Finanzmathematik by Ludger Rüschendorf, published by Springer-Verlag GmbH Germany, part of Springer Nature in 2020. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com) and in a subsequent editing, improved by the author. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors.

This book is devoted to a domain of highest industrial and scienti?c interest, the complexity. The complexity understanding and management will be a main source of e?ciency and prosperity for the next decades. Complex systems areassembliesof multiple subsystemsand arecharact- ized by emergent behavior that results by nonlinear interactions among the subsystems at multiple levels of organization. Evolvability that is the ability to evolve is the method to confront and surpass the successive boundaries of complexity. Evolvability is not biological but should be considered here in the sense that the corresponding systems have, at di?erent levels, charact- istics that are naturally associated to the living systems. The signi?cance of the complexity and the phenomena of emergence are highlighted in the ?rst chapterofthe book.Theimplicationofconcepts aslevelofreality, circularity and closure for evolvable systems is evaluated. The second chapter of the book exposes the methodology to analyze and manage complex systems. The polystochastic models, PSMs, are the cons- ered mathematical tools. PSMs characterize systems emerging when several stochastic processes occurring at di?erent conditioning levels, are capable to interact with each other, resulting in qualitatively new processes and s- tems. Innovative are the higher categories approach and the introduction of apartialdi?erentialmodelfor multiple levelsmodeling.This imposes making use of appropriate notions of time, space, probabilities and entropy. Categorytheoryistheformalismcapabletooutlinethegeneralframework, shared by the functional organization of biological organisms, of cognitive systems, by the operational structure of evolvable technologies and devices and after all by the scienti?c and engineering methods

Thiscollectionofproblemsisplannedasatextbookforuniversitycoursesinthe theoryofstochasticprocessesandrelatedspecialcourses. Theproblemsinthebook haveawidespectrumofthelevelofdif cultyandcanbeusefulforreaderswith variouslevelsofmasteringinthetheoryofstochasticprocesses. Togetherwithte- nicalandillustrativeproblemsintendedforbeginners,thebookcontainsanumber ofproblemsoftheoreticalnaturethatcanbeusefulforstudentsandundergraduate studentsthatpursueadvancedstudiesinthetheoryofstochasticprocessesandits- plications. Amongothers,theimportantaimofthebookistoprovideateachingstaff anef cienttoolforpreparingseminarstudies,tests,andexamsconcerninguniversity coursesinthetheoryofstochasticprocessesandrelatedtopics. Whilecomposingthe book,theauthorshavepartiallyusedthecollectionsofproblemsinprobabilityt- ory[16,65,75,83]. Also,someexercisesandproblemsfromthemonographsand textbooks[4,9,19,22,82]wereused. Atthesametime,alargepartofourproblem bookcontainsoriginalmaterial. Thebookisorganizedasfollows. Theproblemsarecollectedintochapters,each chapterbeingdevotedtoacertaintopic. Atthebeginningofeachchapter,theth- reticalgroundsforthecorrespondingtopicaregivenbrie ytogetherwiththelistof bibliography,whichthereadercanuseinordertostudythistopicinmoredetail. For themostoftheproblems,eitherhintsorcompletesolutions(oranswers)aregiven, andsomeoftheproblemsareprovidedwithbothhintsandsolutions(answers). H- ever,theauthorsdonotrecommendthatareaderusethehintssystematically,because solvingaproblemwithoutassistanceismuchmoreusefulthanusingaready-made idea. Somestatementsthathaveaparticulartheoreticalinterestareformulatedon theoreticalgrounds,andtheirproofsareformulatedasproblemsforthereader. Such problemsaresuppliedwitheithercompletesolutionsordetailedhints. Inordertoworkwiththeproblembookef ciently,areadershouldbeacquainted withprobabilitytheory,calculus,andmeasuretheorywithinthescopeofresp- tiveuniversity courses. Standard notions, suchas random variable, measurability, independence, Lebesgue measure and integral, and so on are used without ad- tionaldiscussion. Allthenewnotionsandstatementsrequiredforsolvingthepr- lemsaregiveneitherontheoreticalgroundsorintheformulationsoftheproblems vii viii Preface straightforwardly. However,sometimesanotionisusedinthetextbeforeitsformal de nition. Forinstance,theWienerandPoissonprocessesareprocesseswithin- pendentincrementsandthusareformallyintroducedinaTheoreticalgroundsfor Chapter5,buttheseprocessesareusedwidelyintheproblemsofChapters2to4. Theauthorsrecommendthatareaderwhocomestoanunknownnotionorobject usetheIndexinorderto ndthecorrespondingformalde nition. Thesamerec- mendationconcernssomestandardabbreviationsandsymbolslistedattheendofthe book. Someproblemsinthebookformcycles:solutionstooneofthemaregrounded onstatementsofothersoronauxiliaryconstructionsdescribedinsomepreceding solutions. Sometimes,onthecontrary,itisproposedtoprovethesamestatement withindifferentproblemsusingessentiallydifferenttechniques. Theauthorsrec- mendareaderpayspeci cattentiontothesefruitfulinternallinksbetweenvarious topicsofthetheoryofstochasticprocesses. Everypartofthebookwascomposedsubstantiallybyoneauthor. Chapters1-6, and16arecomposedbyA. Kulik,Chapters7,12-15,18,and19byYu. Mishura, Chapters 8-10 by A. Pilipenko, Chapter 17 by A. Kukush, and Chapter 20 by D. Gusak. Chapter11waspreparedjointlybyD. GusakandA. Pilipenko. Atthe sametime,everyauthorhasmadeacontributiontootherpartsofthebookbyprop- ingseparateproblemsorcyclesofproblems,improvingpreliminaryversionsoft- oreticalgrounds,andeditingthe naltext. The authors would like to express their deep gratitude to M. Portenko and A. Ivanovfortheircarefulreadingofapreliminaryversionofthebookandva- ablecommentsthatledtosigni cantimprovementofthetext. Theauthorsarealso gratefultoT. Yakovenko,G. Shevchenko,O. Soloveyko, Yu. Kartashov, Yu. K- menko,A. Malenko,andN. Ryabovafortheirassistanceintranslation,preparing lesandpictures,andcomposingthesubjectindexandreferences. Thetheoryofstochasticprocessesisanextendeddiscipline,andtheauthors- derstandthattheproblembookinitscurrentformmaycausecriticalremarksfrom readers,concerningeitherthestructureofthebookorthecontentofseparatech- ters. Whilepublishingtheproblembookinitscurrentform,theauthorsareopenfor remarks,comments,andpropositions,andexpressinadvancetheirgratitudetoall theircorrespondents. Kyiv DmytroGusak December2008 AlexanderKukush AlexeyKulik YuliyaMishura AndreyPilipenko Contents 1 De?nition of stochastic process. Cylinder?-algebra, ?nite-dimensional distributions, the Kolmogorov theorem...1 Theoreticalgrounds ...1 Bibliography...3 Problems...3 Hints...7 AnswersandSolutions...9 2 Characteristics of a stochastic process. Mean and covariance functions. Characteristic functions...11 Theoreticalgrounds ...11 Bibliography...13 Problems...13 Hints...16 AnswersandSolutions...17 3 Trajectories. Modi?cations. Filtrations...21 Theoreticalgrounds ...21 Bibliography...24 Problems...24 Hints...29 AnswersandSolutions...31 4 Continuity. Differentiability. Integrability...33 Theoreticalgrounds ...33 Bibliography...34 Problems...34 Hints...38 AnswersandSolutions...40 ix x Contents 5 Stochastic processes with independent increments. Wiener and Poisson processes. Poisson point measures...

Univariate statistical analysis is concerned with techniques for the analysis of a single random variable. This book is about applied multivariate analysis. It was written to p- vide students and researchers with an introduction to statistical techniques for the ana- sis of continuous quantitative measurements on several random variables simultaneously. While quantitative measurements may be obtained from any population, the material in this text is primarily concerned with techniques useful for the analysis of continuous obser- tions from multivariate normal populations with linear structure. While several multivariate methods are extensions of univariate procedures, a unique feature of multivariate data an- ysis techniques is their ability to control experimental error at an exact nominal level and to provide information on the covariance structure of the data. These features tend to enhance statistical inference, making multivariate data analysis superior to univariate analysis. While in a previous edition of my textbook on multivariate analysis, I tried to precede a multivariate method with a corresponding univariate procedure when applicable, I have not taken this approach here. Instead, it is assumed that the reader has taken basic courses in multiple linear regression, analysis of variance, and experimental design. While students may be familiar with vector spaces and matrices, important results essential to multivariate analysis are reviewed in Chapter 2. I have avoided the use of calculus in this text.

This volume provides a modern introduction to stochastic geometry, random fields and spatial statistics at a (post)graduate level. It is focused on asymptotic methods in geometric probability including weak and strong limit theorems for random spatial structures (point processes, sets, graphs, fields) with applications to statistics. Written as a contributed volume of lecture notes, it will be useful not only for students but also for lecturers and researchers interested in geometric probability and related subjects.

The chapters in this volume, written by international experts from different fields of mathematics, are devoted to honoring George Isac, a renowned mathematician. These contributions focus on recent developments in complementarity theory, variational principles, stability theory of functional equations, nonsmooth optimization, and several other important topics at the forefront of nonlinear analysis and optimization.

During the of Fall 1991, The Centre de Recerca Matematica, a research institute sponsored by the Institut d'Estudis Catalans, devoted a quarter to the study of stochastic analysis. Prominent workers in this field visited the Center from all over the world for periods ranging from a few days to several weeks. To take advantage of the presence in Barcelona of so many special ists in stochastic analysis, we organized a workshop on the subject in Sant Feliu de Guixols (Girona) that provided an opportunity for them to ex change information and ideas about their current work. Topics discussed included: Analysis on the Wiener space, Anticipating Stochastic Calculus and its Applications, Correlation Inequalities, Stochastic Flows, Reflected Semimartingales, and others. This volume contains a refereed selection of contributions from some of the participants in this workshop. We are deeply indebted to the authors of the articles for these exposi tions of their valuable research contributions. We also would like to thank all the referees for their helpful advice in making the volume a reflection of the dynamic interchange that characterized the workshop. The success of the Seminar was due essentially to the enthusiasm and stimulating discus sions of all the participants in an informal and pleasant atmosphere. To all of them our warm gratitude."

This volume contains eight artieles by five authors. The common theme is indicated by the title, "Investigations in the Theory of Stochastic Processes. " The artiele by Vershik and Sudakov is a summary of severallectures delivered by the authors on the general aspects of measure theory in linear spaces. The main attention here is focused on the development of certain general concepts within whose framework are studied both the foundations of the theory of random processes as weIl as more specialized probIems. The published summary does not presume prior acquaintance with the subj eet. The group of artieles by Ibragimov and Solev is devoted to questions connected with the regular- ity of stationary random processes. Ibragimov, in "Conditions for the Complete Regularity of Station- ary Processes with Continuous Time," derives suffi"ient conditions for the complete regularity of a stationary random process with continuous time. The case of a rapid decrease of the regularity coef- ficient (in power or exponential form) is investigated. In his other artiele, "Complete Regularity of Generalized Stationary Random Processes," he shows that the problem of investigating the conditions for the complete regularity of generalized stationary random processes reduces to the analogous problem for general stationary random processes.

A NATO Advanced Research Workshop on Classical and Modern Potential The- ory and Applications was held at the Chateau de Bonas, France, during the last week of July 1993. The workshop was organized by the Co-Directors M. Goldstein (Ari- zona) and K. GowriSankaran (Montreal). The other members of the organizing committee were J. Bliedtner (Frankfurt), D. Feyel (Paris), W. K. Hayman (York, England) and I. Netuka (Praha). The objective of the workshop was to bring to- gether the researchers at the forefront of the aspects of the Potential Theory for a meaningful dialogue and for positive interaction amongst the mathematicians prac- tising different aspects of the theory and its applications. Fifty one mathematicians participated in the workshop. The workshop covered a fair representation of the classical aspects of the theory covering topics such as approximations, radial be- haviour, value distributions of meromorphic functions and the modern Potential theory including axiomatic developments, probabilistic theories, studies on infinite dimensional Wiener spaces, solutions of powers of Laplacian and other second order partial differential equations. There were keynote addresses delivered by D. Armitage (Belfast), N. Bouleau (Paris), A. Eremenko (Purdue), S. J. Gardiner (Dublin), W. Hansen (Bielefeld), W. Hengartner (Laval U. , Quebec), K. Janssen (Dusseldorf), T. Murai (Nagoya), A. de la Pradelle (Paris) and J. M. Wu (Urbana). There were thirty six other invited talks of one half hour duration each.

Many special functions occuring in physics and partial differential equations can be represented by integral transformatIons: the fundamental solutions of many PDE's, Newton-Coulomb potentials, hypergeometric functions, Feynman integrals, initial data of (inverse) tomography problems, etc. The general picture of such transfor- mations is as follows. There is an analytic fibre bundle E --+ T, a differential form w on E, whose restrictions on the fibres are closed, and a family of cycles in these fibres, parametrized by the points of T and depending continuously on these points. Then the integral of the form w along these cycles is a function on the base. The analytic properties of such functions depend on the monodromy action, i.e., on the natural action of the fundamental group of the base in the homology of the fibre: this action on the integration cycles defines the ramification of the analytic continuation of our function. The study of this action (which is a purely topological problem) can answer questions about the analytic behaviour of the integral function, for instance, is this function single-valued or at least algebraic, what are the singular points of this function, and what is its asymptotics close to these points. In this book, we study such analytic properties of three famous classes of func- tions: the volume functions, which appear in the Archimedes-Newton problem on in- tegrable bodies; the Newton-Coulomb potentials, and the Green functions of hyperbolic equations (studied, in particular, in the Hada- mard-Petrovskii-Atiyah-Bott-Garding lacuna theory).

This book provides a systematic and comprehensive account of asymptotic sets and functions from which a broad and useful theory emerges in the areas of optimization and variational inequalities. A variety of motivations leads mathematicians to study questions about attainment of the infimum in a minimization problem and its stability, duality and minmax theorems, convexification of sets and functions, and maximal monotone maps. For each there is the central problem of handling unbounded situations. Such problems arise in theory but also within the development of numerical methods. The book focuses on the notions of asymptotic cones and associated asymptotic functions that provide a natural and unifying framework for the resolution of these types of problems. These notions have been used largely and traditionally in convex analysis, yet these concepts play a prominent and independent role in both convex and nonconvex analysis. This book covers convex and nonconvex problems, offering detailed analysis and techniques that go beyond traditional approaches. The book will serve as a useful reference and self-contained text for researchers and graduate students in the fields of modern optimization theory and nonlinear analysis.

This book contains an introduction to three topics in stochastic control: discrete time stochastic control, i. e. , stochastic dynamic programming (Chapter 1), piecewise - terministic control problems (Chapter 3), and control of Ito diffusions (Chapter 4). The chapters include treatments of optimal stopping problems. An Appendix - calls material from elementary probability theory and gives heuristic explanations of certain more advanced tools in probability theory. The book will hopefully be of interest to students in several ?elds: economics, engineering, operations research, ?nance, business, mathematics. In economics and business administration, graduate students should readily be able to read it, and the mathematical level can be suitable for advanced undergraduates in mathem- ics and science. The prerequisites for reading the book are only a calculus course and a course in elementary probability. (Certain technical comments may demand a slightly better background. ) As this book perhaps (and hopefully) will be read by readers with widely diff- ing backgrounds, some general advice may be useful: Don't be put off if paragraphs, comments, or remarks contain material of a seemingly more technical nature that you don't understand. Just skip such material and continue reading, it will surely not be needed in order to understand the main ideas and results. The presentation avoids the use of measure theory.

Identifying the input-output relationship of a system or discovering the evolutionary law of a signal on the basis of observation data, and applying the constructed mathematical model to predicting, controlling or extracting other useful information constitute a problem that has been drawing a lot of attention from engineering and gaining more and more importance in econo metrics, biology, environmental science and other related areas. Over the last 30-odd years, research on this problem has rapidly developed in various areas under different terms, such as time series analysis, signal processing and system identification. Since the randomness almost always exists in real systems and in observation data, and since the random process is sometimes used to model the uncertainty in systems, it is reasonable to consider the object as a stochastic system. In some applications identification can be carried out off line, but in other cases this is impossible, for example, when the structure or the parameter of the system depends on the sample, or when the system is time-varying. In these cases we have to identify the system on line and to adjust the control in accordance with the model which is supposed to be approaching the true system during the process of identification. This is why there has been an increasing interest in identification and adaptive control for stochastic systems from both theorists and practitioners."

From Markov Jump Processes to Spatial Queues aims to develop a unified theory of spatial queues that yields concrete results for the performance analysis of mobile communication networks. A particular objective is to develop the most natural generalization of existing concepts (e.g. the BMAP) toward the needs of mobile communication networks. To these belong the spatial distribution of batch arrivals and users in the system as well as time-inhomogeneous (e.g. periodic) arrival intensities and user movements.

One of the major recent challenges for the stochastic modelling of communication systems is the emergence of wireless networks, which are used by more and more subscribers today. The main new feature of those, which is not covered by classical queuing theory, clearly is the importance of the user location within the area that is served by the base stations of the network.

In the framework of queuing theory, this opens up the natural extension of classical queuing models towards queues with a structured space in which users are served. The present book is intended to introduce this extension under the name of spatial queues. The main point of view and the general approach will be that of Markov jump processes. We start with a closer look into the theory. Then we present new results for the theory of stochastic processes as well as for classical queuing theory. Finally we introduce the new concepts of spatial Markovian arrival processes and spatial queues.

The main text is divided into three parts. The first part provides a new presentation of the theory of Markov jump processes. We derive a number of new results, especially for time-inhomogeneous processes, which have been neglected too much in the current textbooks on stochastic processes. For the first time, the class of Markov-additive jump processes is analysed in detail. This extends and unifies all Markovian arrival processes that have been proposed up to now (including arrivals for fluid queues) and provides a foundation for the subsequent introduction of spatial Markovian arrival processes.

The second part contains new results for classical queues with BMAP input. These include the first explicit formulae for the distribution of periodic queues. The class of fluid Markovian arrival processes is introduced, and we give statistical estimates for the parameters of a BMAP.

In the third part, the concepts of spatial Markovian arrival processes (abbreviated: SMAPs) and spatial queues are introduced. After that, periodic spatial Markovian queues are analysed as a model for the cells of a wireless communication network.
From Markov Jump Processes to Spatial Queues is intended to reach queuing theorists, researchers in the field of communication systems, as well as engineers with some background in probability theory. Furthermore, it is suitable as a textbook for advanced queuing theory on the graduate or post-graduate level.

Quasiregular Mappings extend quasiconformal theory to the noninjective case.They give a natural and beautiful generalization of the geometric aspects ofthe theory of analytic functions of one complex variable to Euclidean n-space or, more generally, to Riemannian n-manifolds. This book is a self-contained exposition of the subject. A braod spectrum of results of both analytic and geometric character are presented, and the methods vary accordingly. The main tools are the variational integral method and the extremal length method, both of which are thoroughly developed here. Reshetnyak's basic theorem on discreteness and openness is used from the beginning, but the proof by means of variational integrals is postponed until near the end. Thus, the method of extremal length is being used at an early stage and leads, among other things, to geometric proofs of Picard-type theorems and a defect relation, which are some of the high points of the present book.

This book discusses a variety of problems which are usually treated in a second course on the theory of functions of one complex variable, the level being gauged for graduate students. It treats several topics in geometric function theory as well as potential theory in the plane, covering in particular: conformal equivalence for simply connected regions, conformal equivalence for finitely connected regions, analytic covering maps, de Branges' proof of the Bieberbach conjecture, harmonic functions, Hardy spaces on the disk, potential theory in the plane. A knowledge of integration theory and functional analysis is assumed.

This self-contained text/reference provides a basic foundation for practitioners, researchers, and students interested in any of the diverse areas of multiscale (geo)potential theory. New mathematical methods are developed enabling the gravitational potential of a planetary body to be modeled using a continuous flow of observations from land or satellite devices. Harmonic wavelets methods are introduced, as well as fast computational schemes and various numerical test examples. Presented are multiscale approaches for numerous geoscientific problems, including geoidal determination, magnetic field reconstruction, deformation analysis, and density variation modelling With exercises at the end of each chapter, the book may be used as a textbook for graduate-level courses in geomathematics, applied mathematics, and geophysics. The work is also an up-to-date reference text for geoscientists, applied mathematicians, and engineers.

The present volume gives a systematic treatment of potential functions. It takes its origin in two courses, one elementary and one advanced, which the author has given at intervals during the last ten years, and has a two-fold purpose: first, to serve as an introduction for students whose attainments in the Calculus include some knowledge of partial derivatives and multiple and line integrals; and secondly, to provide the reader with the fundamentals of the subject, so that he may proceed immediately to the applications, or to the periodical literature of the day. It is inherent in the nature of the subject that physical intuition and illustration be appealed to freely, and this has been done. However, that the book may present sound ideals to the student, and in order also serve the mathematician, both for purposes of reference and as a basis for further developments, the proofs have been given by rigorous methods. This has led, at a number of points, to results either not found elsewhere, or not readily accessible. Thus, Chapter IV contains a proof for the general regular region of the divergence theorem (Gauss', or Green's theorem) on the reduction of volume to surface integrals. The treatment of the fundamental existence theorems in Chapter XI by means of integral equations meets squarely the difficulties incident to .the discontinuity of the kernel, and the same chapter gives an account of the most recent developments with respect to the Dirichlet problem."

Fast technological development produces systems of ever-increasing complex ity. The demand for reliable functioning of these systems has become more and more important. Thus, there is a need for highly reliable technical devices and systems, for monitoring and controlling their functioning and for planning maintenance and corrective actions to fulfill given requirements considering eco nomic limitations. These tasks reflect the wide field of engineering activities that are accompa nied by and based on a wide range of stochastical models. The book presents the main contributions to a workshop on Stochastic Models of Reliability, Qual ity, and Safety held in Schierke near Magdeburg, Germany. This workshop was part of a series of meetings that take place every two years organized by the Society of Reliability, Quality and Safety. The basic idea of these workshops is to bring together theorists, applied statisticians, and practitioners to exchange experiences and ideas of common interest. The book contains recent results in reliability and related fields. The presentation aims at making at least a part of the results accessible to engineers."

This volume contains the contributions of the participants to the Oslo- Silivri Workshop on Stochastic Analysis, held in Silivri, from July 18 to July 29, at the Nazlm Terzioglu Graduate Research Center of Istanbul University. 1994, There were three lectures: * Mathematical Theory 0/ Communication Networks by V. Anantharam, * State-Space Models 0/ the Term Structure o/Interest Rates, by D. Duffie, * Theory 0/ Capacity on the Wiener Space, by F. Hirsch. The main lectures are presented at the beginning of the volume. The contributing papers cover different domains varying from random fields to dis- tributions on infinite dimensional spaces. We would like to thank the following organizations for their financial sup- port: * VISTA, a research cooperation between the Norwegian Academy of Scineces and Letters and Den Norske Stats Oljeselskap A. S. (Statsoil). * Ecole Nationale Superieure des Telecommunications de Paris. In the summer of 1994 we lost our dear friend and colleague ALBERT BADRIKIAN. We are dedicating this volume to his memory. H. Korezlioglu, B. 0ksendal, A. S. Ustunel MATHEMATICAL THEORY OF COMMUNICATION NETWORKS VENKAT ANANTHARAM * EECS DEPARTMENT UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720 [email protected] Abstract We describe so me recent advances in the mathematical theory of com- munication networks.

On behalf of those of us who in various ways have con tributed to this volume, and on behalf of all of his colleagues, students and friends throughout the world-wide scientific com munity, we dedicate this volume to Gopinath Kallianpur as a tribute to his work and in appreciation for the insights which he has so graciously and generously offered, and continues to offer, to all of us. Stochastic Processes contains 41 articles related to and frequently influ enced by Kallianpur's work. We regret that space considerations prevented us from including contributions from his numerous colleagues (at North Carolina, lSI, Minnesota, Michigan), former students, co-authors and other eminent scientists whose work is akin to Kallianpur's. This would have taken several more volumes All articles have been refereed, and for their valuable assistance in this we thank many of the contributing authors, as well as: R. Bradley, M.H.A. Davis, R. Davis, J. Hawkins, J. Horowitz, C. Houdre, N.C. Jain, C. Ji, P. Kokoszka, T. Kurtz, K.S. Lau, W. Linde, D. Monrad, D. Stroook, D. Surgailis and S. Yakowitz. We also thank June Maxwell for editorial assistance, Peggy Ravitch for help with the production of the volume, and Lisa Brooks for secretarial assistance. Finally, we are indebted to Dr. Martin Gilchrist, the Statistics editor, and the Springer editorial board for their excellent cooperation and enthusiastic support throughout this project."

Intended for a second course in stationary processes, Stationary Stochastic Processes: Theory and Applications presents the theory behind the field's widely scattered applications in engineering and science. In addition, it reviews sample function properties and spectral representations for stationary processes and fields, including a portion on stationary point processes. Features Presents and illustrates the fundamental correlation and spectral methods for stochastic processes and random fields Explains how the basic theory is used in special applications like detection theory and signal processing, spatial statistics, and reliability Motivates mathematical theory from a statistical model-building viewpoint Introduces a selection of special topics, including extreme value theory, filter theory, long-range dependence, and point processes Provides more than 100 exercises with hints to solutions and selected full solutions This book covers key topics such as ergodicity, crossing problems, and extremes, and opens the doors to a selection of special topics, like extreme value theory, filter theory, long-range dependence, and point processes, and includes many exercises and examples to illustrate the theory. Precise in mathematical details without being pedantic, Stationary Stochastic Processes: Theory and Applications is for the student with some experience with stochastic processes and a desire for deeper understanding without getting bogged down in abstract mathematics.