Stochastics Books

Book SynopsisThis open access book presents the key aspects of statistics in Wasserstein spaces, i.e. statistics in the space of probability measures when endowed with the geometry of optimal transportation. Further to reviewing state-of-the-art aspects, it also provides an accessible introduction to the fundamentals of this current topic, as well as an overview that will serve as an invitation and catalyst for further research. Statistics in Wasserstein spaces represents an emerging topic in mathematical statistics, situated at the interface between functional data analysis (where the data are functions, thus lying in infinite dimensional Hilbert space) and non-Euclidean statistics (where the data satisfy nonlinear constraints, thus lying on non-Euclidean manifolds). The Wasserstein space provides the natural mathematical formalism to describe data collections that are best modeled as random measures on Euclidean space (e.g. images and point processes). Such random measures carry the infinite dimensional traits of functional data, but are intrinsically nonlinear due to positivity and integrability restrictions. Indeed, their dominating statistical variation arises through random deformations of an underlying template, a theme that is pursued in depth in this monograph.Table of ContentsOptimal transportation.- The Wasserstein space.- Fréchet means in the Wasserstein space.- Phase variation and Fréchet means.- Construction of Fréchet means and multicouplings.

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Book SynopsisThis book gathers the most essential results, including recent ones, on linear-quadratic optimal control problems, which represent an important aspect of stochastic control. It presents results for two-player differential games and mean-field optimal control problems in the context of finite and infinite horizon problems, and discusses a number of new and interesting issues. Further, the book identifies, for the first time, the interconnections between the existence of open-loop and closed-loop Nash equilibria, solvability of the optimality system, and solvability of the associated Riccati equation, and also explores the open-loop solvability of mean-filed linear-quadratic optimal control problems. Although the content is largely self-contained, readers should have a basic grasp of linear algebra, functional analysis and stochastic ordinary differential equations. The book is mainly intended for senior undergraduate and graduate students majoring in applied mathematics who are interested in stochastic control theory. However, it will also appeal to researchers in other related areas, such as engineering, management, finance/economics and the social sciences.Table of Contents1.- Some Elements of Linear-Quadratic Optimal Controls.- 2. Linear-Quadratic Two-Person Differential Games.- 3. Mean-Field Linear-Quadratic Optimal Controls.

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Book SynopsisThis volume provides an introduction to the theory of Mean Field Games, suggested by J.-M. Lasry and P.-L. Lions in 2006 as a mean-field model for Nash equilibria in the strategic interaction of a large number of agents. Besides giving an accessible presentation of the main features of mean-field game theory, the volume offers an overview of recent developments which explore several important directions: from partial differential equations to stochastic analysis, from the calculus of variations to modeling and aspects related to numerical methods. Arising from the CIME Summer School "Mean Field Games" held in Cetraro in 2019, this book collects together lecture notes prepared by Y. Achdou (with M. Laurière), P. Cardaliaguet, F. Delarue, A. Porretta and F. Santambrogio.These notes will be valuable for researchers and advanced graduate students who wish to approach this theory and explore its connections with several different fields in mathematics.Table of Contents- An Introduction to Mean Field Game Theory. - Lecture Notes on Variational Mean Field Games. - Master Equation for Finite State Mean Field Games with Additive Common Noise. - Mean Field Games and Applications: Numerical Aspects.

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Book SynopsisThis book provides an introduction to the theory and applications of point processes, both in time and in space. Presenting the two components of point process calculus, the martingale calculus and the Palm calculus, it aims to develop the computational skills needed for the study of stochastic models involving point processes, providing enough of the general theory for the reader to reach a technical level sufficient for most applications. Classical and not-so-classical models are examined in detail, including Poisson–Cox, renewal, cluster and branching (Kerstan–Hawkes) point processes.The applications covered in this text (queueing, information theory, stochastic geometry and signal analysis) have been chosen not only for their intrinsic interest but also because they illustrate the theory. Written in a rigorous but not overly abstract style, the book will be accessible to earnest beginners with a basic training in probability but will also interest upper graduate students and experienced researchers.Table of ContentsIntroduction.- Generalities.- Poisson Process on the Line.- Spatial Poisson Processes.- Renewal and Regenerative Processes.- Point Processes with a Stochastic Intensity.- Exvisible Intensity of Finite Point Processes.- Palm Probability on the Line.- Palm Probability in Space.- The Power Spectral Measure.- Information Content of Point Processes.- Point Processes in Queueing.- Hawkes Point Processes.- Appendices.- Bibliography.- Index.

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Book SynopsisThis book provides an in-depth account of modern methods used to bound the supremum of stochastic processes. Starting from first principles, it takes the reader to the frontier of current research. This second edition has been completely rewritten, offering substantial improvements to the exposition and simplified proofs, as well as new results.The book starts with a thorough account of the generic chaining, a remarkably simple and powerful method to bound a stochastic process that should belong to every probabilist’s toolkit. The effectiveness of the scheme is demonstrated by the characterization of sample boundedness of Gaussian processes. Much of the book is devoted to exploring the wealth of ideas and results generated by thirty years of efforts to extend this result to more general classes of processes, culminating in the recent solution of several key conjectures.A large part of this unique book is devoted to the author’s influential work. While many of the results presented are rather advanced, others bear on the very foundations of probability theory. In addition to providing an invaluable reference for researchers, the book should therefore also be of interest to a wide range of readers.Trade Review“The book includes a rich collection of exercises that will allow the reader to gain skills for a better understanding. The book is then suitable as a textbook for an advanced course. … The systematic and deep treatment of the subject under study makes the book a good reference for the specialist.” (Erick Treviño-Aguilar, Mathematical Reviews, March, 2023)Table of Contents1. What is This Book About? Part I The Generic Chaining.- 2 Gaussian Processes and the Generic Chaining.- 3 Trees and Other Measures of Size.- 4 Matching Theorems.- Part II Some Dreams Come True.- 5 Warming Up with p-Stable Processes.- 6 Bernoulli Processes.- 7 Random Fourier Series and Trigonometric Sums.- 8 Partitioning Scheme and Families of Distances.- 9 Peaky Part of Functions.- 10 Proof of the Bernoulli Conjecture.- 11 Random Series of Functions.- 12 Infinitely Divisible Processes.- 13 Unfulfilled Dreams.- Part III Practicing.- 14 Empirical Processes, II.- 15 Gaussian Chaos.- 16 Convergence of Orthogonal Series; Majorizing Measures.- 17 Shor's Matching Theorem.- 18 The Ultimate Matching Theorem in Dimension Three.- 19 Application to Banach Space Theory.- A Discrepancy for Convex Sets.- B Some Deterministic Arguments.- C Classical View of Infinitely Divisible Processes.- D Reading Suggestions.- E Research Directions.- F Solutions of Selected Exercises.- G Comparison with the First Edition.- References.- Index.

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Book SynopsisThis textbook presents the basics of probability and statistical estimation, with a view to applications. The didactic presentation follows a path of increasing complexity with a constant concern for pedagogy, from the most classical formulas of probability theory to the asymptotics of independent random sequences and an introduction to inferential statistics. The necessary basics on measure theory are included to ensure the book is self-contained. Illustrations are provided from many applied fields, including information theory and reliability theory. Numerous examples and exercises in each chapter, all with solutions, add to the main content of the book.Written in an accessible yet rigorous style, the book is addressed to advanced undergraduate students in mathematics and graduate students in applied mathematics and statistics. It will also appeal to students and researchers in other disciplines, including computer science, engineering, biology, physics and economics, who are interested in a pragmatic introduction to the probability modeling of random phenomena.Table of Contents- 1. Events and Probability Spaces. - 2. Random Variables. - 3. Random Vectors. - 4. Random Sequences. - 5. Introduction to Statistics.

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Book SynopsisThis book is a translation of the third edition of the well accepted German textbook 'Stochastik', which presents the fundamental ideas and results of both probability theory and statistics, and comprises the material of a one-year course. The stochastic concepts, models and methods are motivated by examples and problems and then developed and analysed systematically.Trade Review"The book can be used by undergraduate mathematics majors but also by science and engeneering students who wish not only to apply probability and statistics but also to understand how the methods work."Vladimir P. Kurenok in: Mathematical Reviews 2009b "The book is well-written and mathematically oriented students and researchers will certainly find that it provides a high level introduction to probability theory and mathematical statistics."In: EMS Newsletter 9/2008

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Book SynopsisThe present volume is an extensive monograph on the analytic and geometric aspects of Markov diffusion operators. It focuses on the geometric curvature properties of the underlying structure in order to study convergence to equilibrium, spectral bounds, functional inequalities such as Poincaré, Sobolev or logarithmic Sobolev inequalities, and various bounds on solutions of evolution equations. At the same time, it covers a large class of evolution and partial differential equations. The book is intended to serve as an introduction to the subject and to be accessible for beginning and advanced scientists and non-specialists. Simultaneously, it covers a wide range of results and techniques from the early developments in the mid-eighties to the latest achievements. As such, students and researchers interested in the modern aspects of Markov diffusion operators and semigroups and their connections to analytic functional inequalities, probabilistic convergence to equilibrium and geometric curvature will find it especially useful. Selected chapters can also be used for advanced courses on the topic.Trade Review“The book is friendly written and is of a rich content. With simple examples, main ideas of the study are clearly explained and naturally extended to a general framework, so that main progress in the field made for the past decades is presented in a smooth way. The monograph is undoubtedly a significant reference for further development of diffusion semigroups and related topics.” (Feng-Yu Wang, zbMATH 1376.60002, 2018)“It is extremely rich. It is more original and inspirational than a treatise. One can use it and benefit from it in many ways: as a reference book, as an inspiration source, by focusing on a property or on an example. … From the beginning to the end, this book definitely has a strong personality and a characteristic taste. … anybody who wants to explore analytic, probabilistic or geometric properties of Markov semigroups to have a look at it first.” (Thierry Coulhon, Jahresbericht der Deutschen Math-Vereinigung, Vol. 119, 2017)“This impressive monograph is about an important and highly active area that straddles the fertile land occupied by both probability and analysis. … It is written with great clarity and style, and was clearly a labour of love for the authors. I am convinced that it will be a valuable resource for researchers in analysis and probability for many years to come.” (David Applebaum, The Mathematical Gazette, Vol. 100 (548), July, 2016)Table of ContentsIntroduction.- Part I Markov semigroups, basics and examples: 1.Markov semigroups.- 2.Model examples.- 3.General setting.- Part II Three model functional inequalities: 4.Poincaré inequalities.- 5.Logarithmic Sobolev inequalities.- 6.Sobolev inequalities.- Part III Related functional, isoperimetric and transportation inequalities: 7.Generalized functional inequalities.- 8.Capacity and isoperimetry-type inequalities.- 9.Optimal transportation and functional inequalities.- Part IV Appendices: A.Semigroups of bounded operators on a Banach space.- B.Elements of stochastic calculus.- C.Some basic notions in differential and Riemannian geometry.- Notations and list of symbols.- Bibliography.- Index.

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Book SynopsisA certain curious feature of random objects, introduced by the author as “super concentration,” and two related topics, “chaos” and “multiple valleys,” are highlighted in this book. Although super concentration has established itself as a recognized feature in a number of areas of probability theory in the last twenty years (under a variety of names), the author was the first to discover and explore its connections with chaos and multiple valleys. He achieves a substantial degree of simplification and clarity in the presentation of these findings by using the spectral approach.Understanding the fluctuations of random objects is one of the major goals of probability theory and a whole subfield of probability and analysis, called concentration of measure, is devoted to understanding these fluctuations. This subfield offers a range of tools for computing upper bounds on the orders of fluctuations of very complicated random variables. Usually, concentration of measure is useful when more direct problem-specific approaches fail; as a result, it has massively gained acceptance over the last forty years. And yet, there is a large class of problems in which classical concentration of measure produces suboptimal bounds on the order of fluctuations. Here lies the substantial contribution of this book, which developed from a set of six lectures the author first held at the Cornell Probability Summer School in July 2012.The book is interspersed with a sizable number of open problems for professional mathematicians as well as exercises for graduate students working in the fields of probability theory and mathematical physics. The material is accessible to anyone who has attended a graduate course in probability.Table of ContentsPreface.- 1.Introduction.- 2.Markov semigroups.- 3.Super concentration and chaos.- 4.Multiple valleys.- 5.Talagrand’s method for proving super concentration.- 6.The spectral method for proving super concentration.- 7.Independent flips.- 8.Extremal fields.- 9.Further applications of hypercontractivity.- 10.The interpolation method for proving chaos.- 11.Variance lower bounds.- 12.Dimensions of level sets.- Appendix A. Gaussian random variables.- Appendix B. Hypercontractivity.- Bibliography.- Indices.

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Book SynopsisThis research monograph presents results to researchers in stochastic calculus, forward and backward stochastic differential equations, connections between diffusion processes and second order partial differential equations (PDEs), and financial mathematics. It pays special attention to the relations between SDEs/BSDEs and second order PDEs under minimal regularity assumptions, and also extends those results to equations with multivalued coefficients. The authors present in particular the theory of reflected SDEs in the above mentioned framework and include exercises at the end of each chapter.Stochastic calculus and stochastic differential equations (SDEs) were first introduced by K. Itô in the 1940s, in order to construct the path of diffusion processes (which are continuous time Markov processes with continuous trajectories taking their values in a finite dimensional vector space or manifold), which had been studied from a more analytic point of view by Kolmogorov in the 1930s. Since then, this topic has become an important subject of Mathematics and Applied Mathematics, because of its mathematical richness and its importance for applications in many areas of Physics, Biology, Economics and Finance, where random processes play an increasingly important role. One important aspect is the connection between diffusion processes and linear partial differential equations of second order, which is in particular the basis for Monte Carlo numerical methods for linear PDEs. Since the pioneering work of Peng and Pardoux in the early 1990s, a new type of SDEs called backward stochastic differential equations (BSDEs) has emerged. The two main reasons why this new class of equations is important are the connection between BSDEs and semilinear PDEs, and the fact that BSDEs constitute a natural generalization of the famous Black and Scholes model from Mathematical Finance, and thus offer a natural mathematical framework for the formulation of many new models in Finance.Trade Review“This 668-page magnum opus of stochastic ODEs and PDEs belongs on the shelf of every researcher in these areas, as well as any mathematician or scientist who wants to learn more about the subject. … my opinion is that this book accomplished a Herculean task of making an arguably technical subject that is daunting to a beginner accessible. This book wants to be read!” (Mark A. McKibben, Mathematical Reviews, April, 2016)“The present monograph gives a rather complete treatment of backward stochastic differential equations as tool for the stochastic interpretation of second order PDEs. As the reader is guided from basic knowledge on stochastic analysis through the Itō calculus and the theory of stochastic differential equations to that of the backward equations, the monograph represents in my eyes a precious textbook for Master students, PhD students, but also specialists in this domain.” (Rainer Buckdahn, zbMATH 1321.60005, 2015)Table of ContentsIntroduction.- Background of Stochastic Analysis.- Ito’s Stochastic Calculus.- Stochastic Differential Equations.- SDE with Multivalued Drift.- Backward SDE.- Annexes.- Bibliography.- Index. ​ ​

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Book SynopsisThis is the revised and enlarged 2nd edition of the authors’ original text, which was intended to be a modest complement to Grenander's fundamental memoir on stochastic processes and related inference theory. The present volume gives a substantial account of regression analysis, both for stochastic processes and measures, and includes recent material on Ridge regression with some unexpected applications, for example in econometrics.The first three chapters can be used for a quarter or semester graduate course on inference on stochastic processes. The remaining chapters provide more advanced material on stochastic analysis suitable for graduate seminars and discussions, leading to dissertation or research work. In general, the book will be of interest to researchers in probability theory, mathematical statistics and electrical and information theory.Trade Review“A wonderful text with a very high pedagogical and scientific quality, on inference theory in stochastic processes, important for researchers in probability theory, mathematical statistics and electrical and information theory.” (Prof. Dr. Manuel Alberto M. Ferreira, Acta Scientiae et Intellectus, Vol. 2 (1), 2016)“This book is the revised and enlarged edition of the author's original text … . The book is well written and will be of interest for researchers in probability theory and mathematical statistics.” (N. G. Gamkrelidze, zbMATH 1341.62036, 2016)Table of Contents1.Introduction and Preliminaries.- 2.Some Principles of Hypothesis Testing.- 3.Parameter Estimation and Asymptotics.- 4.Inferences for Classes of Processes.- 5.Likelihood Ratios for Processes.- 6.Sampling Methods for Processes.- 7.More on Stochastic Inference.- 8.Prediction and Filtering of Processes.- 9.Nonparametric Estimation for Processes.- Bibliography.- Index.

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Book SynopsisConsidering Poisson random measures as the driving sources for stochastic (partial) differential equations allows us to incorporate jumps and to model sudden, unexpected phenomena. By using such equations the present book introduces a new method for modeling the states of complex systems perturbed by random sources over time, such as interest rates in financial markets or temperature distributions in a specific region. It studies properties of the solutions of the stochastic equations, observing the long-term behavior and the sensitivity of the solutions to changes in the initial data. The authors consider an integration theory of measurable and adapted processes in appropriate Banach spaces as well as the non-Gaussian case, whereas most of the literature only focuses on predictable settings in Hilbert spaces. The book is intended for graduate students and researchers in stochastic (partial) differential equations, mathematical finance and non-linear filtering and assumes a knowledge of the required integration theory, existence and uniqueness results and stability theory. The results will be of particular interest to natural scientists and the finance community. Readers should ideally be familiar with stochastic processes and probability theory in general, as well as functional analysis and in particular the theory of operator semigroups. ​Table of Contents1.Introduction.- 2.Preliminaries.- 3.Stochastic Integrals with Respect to Compensated Poisson Random Measures.- 4.Stochastic Integral Equations in Banach Spaces.- 5.Stochastic Partial Differential Equations in Hilbert Spaces.- 6.Applications.- 7.Stability Theory for Stochastic Semilinear Equations.- A Some Results on compensated Poisson random measures and stochastic integrals.- References.- Index.

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Book SynopsisThis monograph provides a concise presentation of a mathematical approach to metastability, a wide-spread phenomenon in the dynamics of non-linear systems - physical, chemical, biological or economic - subject to the action of temporal random forces typically referred to as noise, based on potential theory of reversible Markov processes. The authors shed new light on the metastability phenomenon as a sequence of visits of the path of the process to different metastable sets, and focuses on the precise analysis of the respective hitting probabilities and hitting times of these sets.The theory is illustrated with many examples, ranging from finite-state Markov chains, finite-dimensional diffusions and stochastic partial differential equations, via mean-field dynamics with and without disorder, to stochastic spin-flip and particle-hop dynamics and probabilistic cellular automata, unveiling the common universal features of these systems with respect to their metastable behaviour. The monograph will serve both as comprehensive introduction and as reference for graduate students and researchers interested in metastability.Trade Review“This monograph gives a complete and detailed account of the most recent techniques developed to obtain a precise mathematical description of the phenomenon of metastability. … The book is well organized and well written, it contains a large amount of fundamental ideas and techniques, and it is an important reference for any researcher interested in the study of long-time behavior of Markov processes and applications to statistical mechanics.” (Jean-Baptiste Bardet, Mathematical Reviews, April, 2017)“No doubt, this is a fundamental book written by well established scientists whose contribution to this area is widely recognized. The book is addressed to readers with serious mathematical background and interests in metastability of stochastic dynamical systems. The books is also an excellent references source.” (Jordan M. Stoyanov, zbMATH 1339.60002, 2016)Table of ContentsPart I Introduction.- 1.Background and motivation.- 2.Aims and scopes.- Part II Markov processes 3.Some basic notions from probability theory.- 4.Markov processes in discrete time.- 5.Markov processes in continuous time.- 6.Large deviations.- 7.Potential theory.- Part III Metastability.- 8.Key definitions and basic properties.- 9.Basic techniques.- Part IV Applications: Diffusions with small noise.- 10.Discrete reversible diffusions.- 11.Diffusion processes with gradient drift.- 12.Stochastic partial differential equations.- Part V Applications: Coarse-graining at positive temperatures.- 13.The Curie-Weiss model.- 14.The Curie-Weiss model with a random magnetic field: discrete distributions.- 15.The Curie-Weiss model with random magnetic field: continuous distributions.- Part VI Applications: Lattice systems in small volumes at low temperatures.- 16.Abstract set-up and metastability in the zero-temperature limit.- 17.Glauber dynamics.- 18.Kawasaki dynamics.- Part VII Applications: Lattice systems in large volumes at low temperatures.- 19.Glauber dynamics.- 20.Kawasaki dynamics.- Part VIII Applications: Lattice systems in small volumes at high densities.- 21.The zero-range process.- Part IX Challenges.- 22.Challenges within metastability.- 23.Challenges beyond metastability.- References.-Glossary.- Index.

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Book SynopsisProceedings of a Conference held in Heidelberg, September 10 - 14, 1984Table of ContentsStochastic spatial processes in biology: A concise historical survey.- Tests for space-time clustering.- Age distributions in birth and death processes.- Critical clustering in the two dimensional voter model.- Measure-valued processes Construction, qualitative behavior and stochastic geometry.- Dual processes in population genetics.- Some peculiar properties of a particle system with sexual reproduction.- Computer simulation of developmental processes in biology: Models for the developing limb.- Asymptotics and spatial growth of branching random fields.- Generation-dependent branching processes with immigration: convergence of distributions.- On a class of infinite particle systems evolving in a random environment.- Percolation processes and dimensionality.- Birth and death processes with killing and applications to parasitic infections.- Limit theorems for multitype branching random walks.- On the reproduction rate of the spatial general epidemic.- Nearest particle systems: Results and open problems.- Neutral models of geographical variation.- Stochastic measure diffusions as models of growth and spread.- L 2 convergence of certain random walks on Z d and related diffusions.- Random fields: Applications in cell biology.- Correlated percolation and repulsive particle systems.

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Book SynopsisThis second BiBoS volume surveys recent developments in the theory of stochastic processes. Particular attention is given to the interaction between mathematics and physics. Main topics include: statistical mechanics, stochastic mechanics, differential geometry, stochastic proesses, quantummechanics, quantum field theory, probability measures, central limit theorems, stochastic differential equations, Dirichlet forms.Table of ContentsJump processes related to the two dimensional dirac equation.- A constructive characterization of radon probability measures on infinite dimensional spaces.- A "Brownian motion" with constant speed.- The semi-martingale approach to the optimal resource allocation in the controlled labour-surplus economy.- A central limit theorem for the laplacian in regions with many small holes.- On dirichlet forms with random data—Recurrence and homogenization.- A nicolai map for supersymmetric quantum mechanics on riemannian manifolds.- Stochastic equations for some Euclidean fields.- Percolation of the two-dimensional ising model.- How do stochastic processes enter into physics?.- Estimates on the difference between succeeding eigenvalues and Lifshitz tails for random Schrödinger operators.- On identification for distributed parameter systems.- Fock space and probability theory.- On a transformation of symmetric markov process and recurrence property.- On absolute continuity of two symmetric diffusion processes.- Collective phenomena in stochastic particle systems.- Boundary problems for stochastic partial differential equations.- Generalized one-sided stable distributions.- Quantum fields, gravitation and thermodynamics.- Self-repellent random walks and polymer measures in two dimensions.- On the uniquness of the markovian self-adjoint extension.- Representations of the group of equivariant loops in SU(N).- Proof of an algebraic central limit theorem by moment generating functions.- Averaging and fluctuations of certain stochastic equations.- Semimartingale with smooth density — The problem of "nodes".

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Book SynopsisThis volume contains detailed, worked-out notes of six main courses given at the Saint-Flour Summer Schools from 1985 to 1987.Table of ContentsLarge deviations and applications.- Applications of non-commutative fourier analysis to probability problems.- Random fields and diffusion processes.- Waves in one-dimensional random media.- Remarks on the point interaction approximation.- Geometric aspects of diffusions on manifolds.- Stochastic mechanics and random fields.

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Book SynopsisSince 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 Itô and McKean.Trade Review"The systematic character of the exposision, which makes from the widely ramified subject matter of the extensive literature a clear, masterly arranged whole, is a particularly valuable feature of this monograph." (Publicationes Mathematicae)Table of ContentsPrerequisites.- 1. The standard BRownian motion.- 1.1. The standard random walk.- 1.2. Passage times for the standard random walk.- 1.3. Hin?in’s proof of the de Moivre-laplace limit theorem.- 1.4. The standard Brownian motion.- 1.5. P. Lévy’s construction.- 1.6. Strict Markov character.- 1.7. Passage times for the standard Brownian motion.- Note l: Homogeneous differential processes with increasing paths.- 1.8. Kolmogorov’s test and the law of the iterated logarithm.- 1.9. P. Lévy’s Hölder condition.- 1.10. Approximating the Brownian motion by a random walk.- 2. Brownian local times.- 2.1. The reflecting Brownian motion.- 2.2. P. Lévy’s local time.- 2.3. Elastic Brownian motion.- 2.4. t+ and down-crossings.- 2.5. T+ as Hausdorff-Besicovitch 1/2-dimensional measure.- Note 1: Submartingales.- Note 2: Hausdorff measure and dimension.- 2.6. Kac’s formula for Brownian functionals.- 2.7. Bessel processes.- 2.8. Standard Brownian local time.- 2.9. BrowNian excursions.- 2.10. Application of the Bessel process to Brownian excursions.- 2.11. A time substitution.- 3. The general 1-dimensional diffusion.- 3.1. Definition.- 3.2. Markov times.- 3.3. Matching numbers.- 3.4. Singular points.- 3.5. Decomposing the general diffusion into simple pieces.- 3.6. Green operators and the space D.- 3.7. Generators.- 3.8. Generators continued.- 3.9. Stopped diffusion.- 4. Generators.- 4.1. A general view.- 4.2. G as local differential operator: conservative non-singular case.- 4.3. G as local differential operator: general non-singular case.- 4.4. A second proof.- 4.5. G at an isolated singular point.- 4.6. Solving G•u = ? u.- 4.7. G as global differential operator: non-singular case.- 4.8. G on the shunts.- 4.9. G as global differential operator: singular case.- 4.10. Passage times.- Note 1: Differential processes with increasing paths.- 4.11. Eigen-differential expansions for Green functions and transition densities.- 4.12. Kolmogorov’s test.- 5. Time changes and killing.- 5.1. Construction of sample paths: a general view.- 5.2. Time changes: Q = R1.- 5.3. Time changes: Q = [0, + ?).- 5.4. Local times.- 5.5. Subordination and chain rule.- 5.6. Killing times.- 5.7. Feller’s Brownian motions.- 5.8. Ikeda’s example.- 5.9. Time substitutions must come from local time integrals.- 5.10. Shunts.- 5.11. Shunts with killing.- 5.12. Creation of mass.- 5.13. A parabolic equation.- 5.14. Explosions.- 5.15. A non-linear parabolic equation.- 6. Local and inverse local times.- 6.1. Local and inverse local times.- 6.2. Lévy measures.- 6.3. t and the intervals of [0, + ?) - ?.- 6.4. A counter example: t and the intervals of [0, + ?) - ?.- 6.5a t and downcrossings.- 6.5b t as Hausdorff measure.- 6.5c t as diffusion.- 6.5d Excursions.- 6.6. Dimension numbers.- 6.7. Comparison tests.- Note 1: Dimension numbers and fractional dimensional capacities.- 6.8. An individual ergodic theorem.- 7. Brownian motion in several dimensions.- 7.1. Diffusion in several dimensions.- 7.2. The standard Brownian motion in several dimensions.- 7.3. Wandering out to ?.- 7.4. Greenian domains and Green functions.- 7.5. Excessive functions.- 7.6. Application to the spectrum of ?/2.- 7.7. Potentials and hitting probabilities.- 7.8. Newtonian capacities.- 7.9. Gauss’s quadratic form.- 7.10. Wiener’s test.- 7.11. Applications of Wiener’s test.- 7.12. Dirichlet problem.- 7.13. Neumann problem.- 7.14. Space-time Brownian motion.- 7.15. Spherical Brownian motion and skew products.- 7.16. Spinning.- 7.17. An individual ergodic theorem for the standard 2-dimensional BROWNian motion.- 7.18. Covering Brownian motions.- 7.19. Diffusions with Brownian hitting probabilities.- 7.20. Right-continuous paths.- 7.21. Riesz potentials.- 8. A general view of diffusion in several dimensions.- 8.1. Similar diffusions.- 8.2. G as differential operator.- 8.3. Time substitutions.- 8.4. Potentials.- 8.5. Boundaries.- 8.6. Elliptic operators.- 8.7. Feller’s little boundary and tail algebras.- List of notations.

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Book SynopsisThis book has been long awaited in the "interacting particle systems" community. Begun by Claude Kipnis before his untimely death, it was completed by Claudio Landim, his most brilliant student and collaborator. It presents the techniques used in the proof of the hydrodynamic behavior of interacting particle systems.Trade Review"Das Buch ist nach Meinung des Rezensenten eine gelungene Einführung in ein interessantes Gebiet der modernen Stochastik und mathematischen Physik und stellt einen fest umrissenen Gegenstand umfassend dar, vor allem den analytisch-methodischen Aspekt. ... Die Beweise sind übersichtlich und gut gegliedert, was das Nachvollziehen der Argumente sehr erleichtert; die didaktische Leistung der Autoren in diesem Punkt ist beeindruckend. Ein sorgfältig zusammengestelltes Literaturverzeichnis von etwa 400 Titeln schließt das Buch ab. Ingesamt ein sehr gut geschriebener und nützlicher Band."DMV Jahresbericht, 103. Band, Heft 3, November 2001Table of Contents1. An Introductory Example: Independent Random Walks.- 2. Some Interacting Particle Systems.- 3. Weak Formulations of Local Equilibrium.- 4. Hydrodynamic Equation of Symmetric Simple Exclusion Processes.- 5. An Example of Reversible Gradient System: Symmetric Zero Range Processes.- 6. The Relative Entropy Method.- 7. Hydrodynamic Limit of Reversible Nongradient Systems.- 8. Hydrodynamic Limit of Asymmetric Attractive Processes.- 9. Conservation of Local Equilibrium for Attractive Systems.- 10. Large Deviations from the Hydrodynamic Limit.- 11. Equilibrium Fluctuations of Reversible Dynamics.- Appendices.- 1. Markov Chains on a Countable Space.- 1.1 Discrete Time Markov Chains.- 1.2 Continuous Time Markov Chains.- 1.3 Kolmogorov’s Equations, Generators.- 1.4 Invariant Measures, Reversibility and Adjoint Processes.- 1.5 Some Martingales in the Context of Markov Processes.- 1.6 Estimates on the Variance of Additive Functionals of Markov Processes.- 1.7 The Feynman-Kac Formula.- 1.8 Relative Entropy.- 1.9 Entropy and Markov Processes.- 1.10 Dirichlet Form.- 1.11 A Maximal Inequality for Reversible Markov Processes.- 2. The Equivalence of Ensembles, Large Deviation Tools and Weak Solutions of Quasi-Linear Differential Equations.- 2.1 Local Central Limit Theorem and Equivalence of Ensembles.- 2.2 On the Local Central Limit Theorem.- 2.3 Remarks on Large Deviations.- 2.4 Weak Solutions of Nonlinear Parabolic Equations.- 2.5 Entropy Solutions of Quasi-Linear Hyperbolic Equations.- 3. Nongradient Tools: Spectral Gap and Closed Forms.- 3.1 On the Spectrum of Reversible Markov Processes.- 3.2 Spectral Gap for Generalized Exclusion Processes.- 3.4 Closed and Exact Forms.- 3.5 Comments and References.- References.

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Book SynopsisCe livre présente une catégorie de modèles probabilistes regroupés sous le nom de réseaux ou systèmes de files d'attente. Ces modèles interviennent dans de nombreuses applications, comme les réseaux de télecommunication ou les réseaux informatiques. Sur le plan théorique ils sont à la source d'une large classe de problèmes : marches aléatoires et diffusions réfléchies, processus ponctuels, etc. Ce livre présente les techniques probabilistes qui permettent d'étudier le comportement qualitatif de ces modèles.Table of ContentsProcessus ponctuels.- Le maximum d'une marche aléatoire.- Réversibilité et équations d'équilibre des réseaux.- La marche aléatoire simple réfléchie.- La file d'attente M/M/infini.- Les files d'attente avec une entrée poissonnienne.- Critères de stabilité.- Méthodes de renormalisation.- Théorie ergodique.- Processus ponctuels stationnaires.- La file d'attente G/G1 FIFO. Annexe.- Loi de Poisson et évènements rares.- Rappels sur les martingales.- Les processus markoviens de sauts.- Convergence en distribution.

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Book SynopsisAlthough three decades have passed since the first publication of this book, it is reprinted now as a result of popular demand. The content remains up-to-date and interesting for many researchers as is shown by the many references to it in current publications. The author is one of the leading experts of the field and gives an authoritative treatment of a subject.Table of ContentsRandom Processes: Markov Times.- Optimal Stopping of Markov Sequences.- Optimal Stopping of Markov Processes.- Some Applications to Problems of Mathematical Statistics.

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Book SynopsisIt was the end of 2005 when our employer, a major European Investment Bank, gave our team the mandate to compute in an accurate way the counterparty credit exposure arising from exotic derivatives traded by the ?rm. As often happens, - posure of products such as, for example, exotic interest-rate, or credit derivatives were modelled under conservative assumptions and credit of?cers were struggling to assess the real risk. We started with a few models written on spreadsheets, t- lored to very speci?c instruments, and soon it became clear that a more systematic approach was needed. So we wrote some tools that could be used for some classes of relatively simple products. A couple of years later we are now in the process of building a system that will be used to trade and hedge counterparty credit ex- sure in an accurate way, for all types of derivative products in all asset classes. We had to overcome problems ranging from modelling in a consistent manner different products booked in different systems and building the appropriate architecture that would allow the computation and pricing of credit exposure for all types of pr- ucts, to ?nding the appropriate management structure across Business, Risk, and IT divisions of the ?rm. In this book we describe some of our experience in modelling counterparty credit exposure, computing credit valuation adjustments, determining appropriate hedges, and building a reliable system.Table of ContentsMethodology.- Modelling Framework.- Simulation Models.- Valuation and Sensitivities.- Architecture and Implementation.- Computational Framework.- Implementation.- Architecture.- Products.- Interest-Rate Products.- Equity, Commodity, Inflation and FX Products.- Credit Derivatives.- Structures.- Hedging and Managing Counterparty Risk.- Counterparty Risk Aggregation and Risk Mitigation.- Combining Market and Credit Risk.- Pricing Counterparty Credit Risk.

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Book Synopsis"This is a magnificent book! Its purpose is to describe in considerable detail a variety of techniques used by probabilists in the investigation of problems concerning Brownian motion....This is THE book for a capable graduate student starting out on research in probability: the effect of working through it is as if the authors are sitting beside one, enthusiastically explaining the theory, presenting further developments as exercises." –BULLETIN OF THE L.M.S.Trade ReviewThis is a magnificent book! Its purpose is to describe in considerable detail a variety of techniques used by probabilists in the investigation of problems concerning Brownian motion. The great strength of Revuz and Yor is the enormous variety of calculations carried out both in the main text and also (by implication) in the exercises. ... This is THE book for a capable graduate student starting out on research in probability: the effect of working through it is as if the authors are sitting beside one, enthusiastically explaining the theory, presenting further developments as exercises, and throwing out challenging remarks about areas awaiting further research..." Bull.L.M.S. 24, 4 (1992) From the reviews of the third edition: "The authors have revised the second edition of their fundamental and impressive monograph on Brownian motion and continuous martingales … . The presentation of this book is unique in the sense that a concise and well-written text is complemented by a long series of detailed exercises. … This third edition contains some additional exercises related to recent advances in the subject. … is a valuable update of this basic reference book, which has been very helpful for researchers and students … ." (David Nualart, Zentralblatt MATH, Vol. 1087, 2006)Table of Contents0. Preliminaries.- I. Introduction.- II. Martingales.- III. Markov Processes.- IV. Stochastic Integration.- V. Representation of Martingales.- VI. Local Times.- VII. Generators and Time Reversal.- VIII. Girsanov’s Theorem and First Applications.- IX. Stochastic Differential Equations.- X. Additive Functionals of Brownian Motion.- XI. Bessel Processes and Ray-Knight Theorems.- XII. Excursions.- XIII. Limit Theorems in Distribution.- §1. Gronwall’s Lemma.- §2. Distributions.- §3. Convex Functions.- §4. Hausdorff Measures and Dimension.- §5. Ergodic Theory.- §6. Probabilities on Function Spaces.- §7. Bessel Functions.- §8. Sturm-Liouville Equation.- Index of Notation.- Index of Terms.- Catalogue.

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Book SynopsisStochastic optimization problems arise in decision-making problems under uncertainty, and find various applications in economics and finance. On the other hand, problems in finance have recently led to new developments in the theory of stochastic control. This volume provides a systematic treatment of stochastic optimization problems applied to finance by presenting the different existing methods: dynamic programming, viscosity solutions, backward stochastic differential equations, and martingale duality methods. The theory is discussed in the context of recent developments in this field, with complete and detailed proofs, and is illustrated by means of concrete examples from the world of finance: portfolio allocation, option hedging, real options, optimal investment, etc. This book is directed towards graduate students and researchers in mathematical finance, and will also benefit applied mathematicians interested in financial applications and practitioners wishing to know more about the use of stochastic optimization methods in finance.Table of ContentsSome elements of stochastic analysis.- Stochastic optimization problems. Examples in finance.- The classical PDE approach to dynamic programming.- The viscosity solutions approach to stochastic control problems.- Optimal switching and free boundary problems.- Backward stochastic differential equations and optimal control.- Martingale and convex duality methods.

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Book SynopsisThis book discloses a fascinating connection between optimal stopping problems in probability and free-boundary problems. It focuses on key examples and the theory of optimal stopping is exposed at its basic principles in discrete and continuous time covering martingale and Markovian methods. Methods of solution explained range from change of time, space, and measure, to more recent ones such as local time-space calculus and nonlinear integral equations. A chapter on stochastic processes makes the material more accessible. The book will appeal to those wishing to master stochastic calculus via fundamental examples. Areas of application include financial mathematics, financial engineering, and mathematical statistics.Table of ContentsOptimal stopping: General facts.- Stochastic processes: A brief review.- Optimal stopping and free-boundary problems.- Methods of solution.- Optimal stopping in stochastic analysis.- Optimal stopping in mathematical statistics.- Optimal stopping in mathematical finance.- Optimal stopping in financial engineering.

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Book SynopsisThe intention of this book is to explain to a mathematician having no previous knowledge in this domain, what "noncommutative probability" is. So the first decision was not to concentrate on a special topic. For different people, the starting points of such a domain may be different. In what concerns this question, different variants are not discussed. One such variant comes from Quantum Physics. The motivations in this book are mainly mathematical; more precisely, they correspond to the desire of developing a probability theory in a new set-up and obtaining results analogous to the classical ones for the newly defined mathematical objects. Also different mathematical foundations of this domain were proposed. This book concentrates on one variant, which may be described as "von Neumann algebras". This is true also for the last chapter, if one looks at its ultimate aim. In the references there are some papers corresponding to other variants; we mention Gudder, S.P. &al (1978). Segal, I.E. (1965) also discusses "basic ideas".Table of ContentsPreface. 1. Central limit theorem on L(H). 2. Probability theory on von Neumann algebras. 3. Free independence. 4. The Clifford algebra. 5. Stochastic integrals. 6. Conditional mean values. 7. Jordan algebras. References. Index.

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Book SynopsisBasics.- The R environment.- Probability and distributions.- Descriptive statistics and graphics.- One- and two-sample tests.- Regression and correlation.- Analysis of variance and the KruskalWallis test.- Tabular data.- Power and the computation of sample size.- Advanced data handling.- Multiple regression.- Linear models.- Logistic regression.- Survival analysis.- Rates and Poisson regression.- Nonlinear curve fitting.Trade ReviewFrom the reviews:TECHNOMETRICS"…extensive, well organized, and well documented…The book is an elegant R companion, suitable for the statistically initiated who want to program their own analyses. For experienced statisticians and data analysts, the book provides a good overview of the basic statistical analysis capabilities of R and presumably prepares readers for later migration to S…The format of this compact book is attractive…The book makes excellent use of fonts and intersperses graphics near the codes that produced them. Output from each procedure is dissected line by line to link R code with the computed result…I can recommend [this book] to its target audience. The author provides an excellent overview of R. I found the wealth of clear examples educational and a practical way to preview both R and S.""The scope of the book, introductory statistics, is a very useful set of methods in parametric and non-parametric statistics up to logistic regression and survival analysis. … Where many constructs in R are very attractive for mathematical oriented users, e.g. matrices, Dalgaard succeeded in convincing me that with little extra effort they can be made very useful to less mathematically oriented people, e.g. by specifying row and column names, and proposing quite attractive ways to specify for example ‘subsets’ of rows and columns." (Dr. H. W. M. Hendriks, Kwantitatieve Methoden, Vol. 72B8, 2003)"R is an Open Source implementation of the well-known S language. It works on multiple computing platforms and can be freely downloaded. R is thus ideally suited for teaching at many levels as well as for practical data analysis and methodological development. This book provides an elementary-level introduction to R, targeting both non-statistician scientists in various fields and students of statistics. … Brief sections introduce the statistical methods before they are used. A supplementary R package can be downloaded and contains the data sets." (Zentralblatt für Didaktik der Mathematik, August, 2004)"This is a nice book on statistical methods and statistical computing in R, a language and environment for statistical computing and graphs: this dialect of the S language is available as free software through internet. … Explanation of statistical methods, together with an interpretation of statistical concepts, is the prevailing style of the text. They are illustrated by plenty of practical examples, all computed using R. This book will be useful for novices in applied statistics or in computing in R." (European Mathematical Society Newsletter, September, 2003)"The book is an elegant R companion, suitable for the statistically initiated who want to program their own analyses. For experienced statisticians and data analysts, the book provides a good overview of the basic statistical analysis capabilities of R … prepares readers for later migration to S. … I can recommend Introductory Statistics With R to its target audience. The author provides an excellent overview of R. I found the wealth of clear examples educational and a practical way to preview both R and S." (Thomas D. Sandry, Technometrics, Vol. 45 (3), 2003)"R is both a statistical computer environment and a programming language designed to perform statistical analysis and to produce adequate corresponding graphics. … The present book is … a very useful guide for introducing a number of basic concepts and techniques necessary to practical statistics, covering both elementary statistics and actual programming in the R language. The book is organized in 12 chapters and three appendices, each chapter ending with a beneficial section of proposed exercises." (Silvia Curteanu, Zentralblatt MATH, Vol. 1006, 2003)From the reviews of the second edition:“This review … roughly cover the introductory topics of a first year statistics course. The Introductory Statistics with R (ISwR) book is targeted for a biometric/medical audience. It covers more topics … like multiple regression and survival analysis and expects the reader to know about basic statistics. … include examples and graphs together with the R code to construct them. … The ISwR book is good for an academic and biometric audience.” (Wolfgang Polasek, Statistical Papers, Vol. 52, 2011)“This is a welcome addition to the new edition that will be appreciated by its users. … The new edition is well written, and the new materials are well incorporated. Like the first edition, this edition will continue to be useful to the target audience, and I can safely recommend it to them.” (Technometrics, Vol. 51 (2), May, 2009)Table of ContentsBasics. - The R environment. - Probability and statistics. - Descriptive statistics and graphics. - One and two sample tests. - Regression and correlation. - ANOVA and Kruskal-Wallis. - Tabular data. - Power and the computation of sample size. - Advanced data handling. - Multiple regression. - Linear models. - Logistic regression. - Survival analysis. - Rates and Poisson regression. - Nonlinear curve-fitting. - Obtaining and installing R and the ISwR package. - Data sets in the ISwR package. - Compendium. - Answers to exercises. - Index.

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Book SynopsisAn introduction to general theories of stochastic processes and modern martingale theory. The volume focuses on consistency, stability and contractivity under geometric invariance in numerical analysis, and discusses problems related to implementation, simulation, variable step size algorithms, and random number generation.Table of ContentsMarkov processes and their applications; semimartingale theory and stochastic calculus; white noise theory; stochastic differential equations and its applications; large deviations and applications; a brief introduction to numerical analysis of (ordinary) stochastic differential equations without tears; stochastic differential games and applications; stability and stabilizing control of stochastic systems; stochastic approximation - theory and applications; stochastic manufacturing systems; optimization by stochastic methods; stochastic control methods in asset pricing.

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Book SynopsisStochastic Modeling for Medical Image Analysis provides a brief introduction to medical imaging, stochastic modeling, and model-guided image analysis.Today, image-guided computer-assisted diagnostics (CAD) faces two basic challenging problems. The first is the computationally feasible and accurate modeling of images from different modalities to obtain clinically useful information. The second is the accurate and fast inferring of meaningful and clinically valid CAD decisions and/or predictions on the basis of model-guided image analysis.To help address this, this book details original stochastic appearance and shape models with computationally feasible and efficient learning techniques for improving the performance of object detection, segmentation, alignment, and analysis in a number of important CAD applications.The book demonstrates accurate descriptions of visual appearances and shapes of the goal objects and their background to help solve aTable of ContentsMedical Imaging Modalities. From Images to Graphical Models. IRF Models: Estimating Marginals. Markov-Gibbs Random Field Models: Estimating Signal Interactions. Applications: Image Alignment. Segmenting Multimodal Images. Segmenting with Deformable Models. Segmenting with Shape and Appearance Priors. Cine Cardiac MRI Analysis. Sizing Cardiac Pathologies.

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Book SynopsisThis book provides a concise introduction to the mathematical foundations of time series analysis, with an emphasis on mathematical clarity. The text is reduced to the essential logical core, mostly using the symbolic language of mathematics, thus enabling readers to very quickly grasp the essential reasoning behind time series analysis. It appeals to anybody wanting to understand time series in a precise, mathematical manner. It is suitable for graduate courses in time series analysis but is equally useful as a reference work for students and researchers alike.Trade Review“‘This book provides a concise introduction to the mathematical foundations of time series analysis, with an emphasis on mathematical clarity. … It appeals to anybody wanting to understand time series in a precise, mathematical manner. It is suitable for graduate courses in time series analysis but is equally useful as a reference work for students and researchers alike.’ … The book can be recommended to all readers, who are interested in this field.” (Ludwig Paditz, zbMath 1414.62001, 2019)“This book is a rigorous, mathematically clear and self-contained and quite complete text on time series analysis, suitable both for graduate courses and as a reference book for researchers and users of stochastic temporal models.” (Nazaré Mendes Lopes, Mathematical Reviews, December, 2018)“Beran (Univ. of Konstanz, Germany) presents the mathematical foundations of time series analysis at a level suitable for advanced graduate students and researchers in statistics. The presentation is extremely concise … . the book gives definitions, theorems, and proofs, along with a few exercises and solutions. … it may be useful to graduate students and researchers as a reference.” (B. Borchers, Choice, Vol. 56 (03), November, 2018)​Table of Contents1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 What is a time series? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Time series versus iid data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Typical assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Fundamental properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 Ergodic property with a constant limit . . . . . . . . . . . . . . . . . . . 52.1.2 Strict Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.3 Weak Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.4 Weak stationarity and Hilbert spaces . . . . . . . . . . . . . . . . . . . . 92.1.5 Ergodic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.1.6 Sufficient conditions for the a.s. ergodic property with a constant limit. . . . . . . . . . . 262.1.7 Sufficient conditions for the L2-ergodic property with a constant limit . .. . . . .. . . 272.2 Specific assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.1 Gaussian processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.2 Linear processes in L2(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.3 Linear processes with E(X2t ) = ∞ . . . . . . . . . . . . . . . . . . . . . . 342.2.4 Multivariate linear processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2.5 Invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.2.6 Restrictions on the dependence structure . . . . . . . . . . . . . . . . . 493 Defining probability measures for time series . . . . . . . . . . . . . . . . . . . . . . 553.1 Finite dimensional distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2 Transformations and equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.3 Conditions on the expected value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.4 Conditions on the autocovariance function . . . . . . . . . . . . . . . . . . . . . . 583.4.1 Positive semidefinite functions . . . . . . . . . . . . . . . . . . . . . . . . . 593.4.2 Spectral distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4.3 Calculation and properties of F and f . . . . . . . . . . . . . . . . .4 Spectral representation of univariate time series . . . . . . . . . . . . . . . . . . . 814.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2 Harmonic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.3 Extension to general processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3.1 Stochastic integrals with respect to Z . . . . . . . . . . . . . . . . . . . . 844.3.2 Existence and definition of Z . . . . . . . . . . . . . . . . . . . . . . . . . . 894.3.3 Interpretation of the spectral representation . . . . . . . . . . . . . . 974.4 Further properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.4.1 Relationship between ReZ and ImZ . . . . . . . . . . . . . . . . . . . . 984.4.2 Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.4.3 Overtones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.4.4 Why are frequencies restricted to the range [-π,π]? . . . . . . . 1004.5 Linear filters and the spectral representation . . . . . . . . . . . . . . . . . . . . 1034.5.1 Effect on the spectral representation . . . . . . . . . . . . . . . . . . . . . 1034.5.2 Elimination of Frequency Bands . . . . . . . . . . . . . . . . . . . . . . . 1075 Spectral representation of real valued vector time series . . . . . . . . . . . . 1095.1 Cross-spectrum and spectral representation . . . . . . . . . . . . . . . . . . . . . 1095.2 Coherence and phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166 Univariate ARMA processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.2 Stationary solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.3 Causal stationary solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.4 Causal invertible stationary solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.5 Autocovariances of ARMA processes . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.5.1 Calculation by integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.5.2 Calculation using the autocovariance generating function . . . 1356.5.3 Calculation using the Wold representation . . . . . . . . . . . . . . . 1386.5.4 Recursive calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.5.5 Asymptotic decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.6 Integrated, seasonal and fractional ARMA and ARIMA processes . . 1476.6.1 Integrated processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476.6.2 Seasonal ARMA processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476.6.3 Fractional ARIMA processes . . . . . . . . . . . . . . . . . . . . . . . . . . 1486.7 Unit roots, spurious correlation, cointegration . . . . . . . . . . . . . . . . . . . 1597 Generalized autoregressive processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.1 Definition of generalized autoregressive processes . . . . . . . . . . . . . . . 1637.2 Stationary solution of generalized autoregressive equations . . . . . . . . 1647.3 Definition of VARMA processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1687.4 Stationary solution of VARMA equations . . . . . . . . . . . . . . . . . . . . . . 1697.5 Definition of GARCH processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1717.6 Stationary solution of GARCH equations . . . . . . . . . . . . . . . . . . . . . . . 1727.7 Definition of ARCH(∞) processes . . . . . . . . . . . . . . . . . . . . .7.8 Stationary solution of ARCH(∞) equations . . . . . . . . . . . . . . . . . . . . . 1778 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1818.1 Best linear prediction given an infinite past . . . . . . . . . . . . . . . . . . . . . 1818.2 Predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1828.3 Construction of the Wold decomposition from f . . . . . . . . . . . . . . . . . 1878.4 Best linear prediction given a finite past . . . . . . . . . . . . . . . . . . . . . . . . 1909 Inference for µ, γ and F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1959.1 Location estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1959.2 Linear regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1979.3 Nonparametric estimation of γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2059.4 Nonparametric estimation of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21110 Parametric estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22710.1 Gaussian and quasi maximum likelihood estimation . . . . . . . . . . . . . . 22710.2 Whittle approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22910.3 Autoregressive approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23210.4 Model choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

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Book SynopsisThe series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.

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Book SynopsisThis volume presents five different methods recently developed to tackle the large scale behavior of highly correlated random systems, such as spin glasses, random polymers, local times and loop soups and random matrices. These methods, presented in a series of lectures delivered within the Jean-Morlet initiative (Spring 2013), play a fundamental role in the current development of probability theory and statistical mechanics. The lectures were: Random Polymers by E. Bolthausen, Spontaneous Replica Symmetry Breaking and Interpolation Methods by F. Guerra, Derrida's Random Energy Models by N. Kistler, Isomorphism Theorems by J. Rosen and Spectral Properties of Wigner Matrices by B. Schlein.This book is the first in a co-edition between the Jean-Morlet Chair at CIRM and the Springer Lecture Notes in Mathematics which aims to collect together courses and lectures on cutting-edge subjects given during the term of the Jean-Morlet Chair, as well as new material produced in its wake. It is targeted at researchers, in particular PhD students and postdocs, working in probability theory and statistical physics.Table of Contents1 Random Polymers.- 2 Spontaneous replica symmetry breaking and interpolation methods for complex statistical mechanics systems.- 3 Derrida’s random energy models: from spin glasses to the extremes of correlated random fields.- 4 Isomorphism Theorems: Markov processes, Gaussian processes and beyond.- 5 Spectral properties of Wigner matrices.

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Book SynopsisEndlich verstehen Sie die ökonomischen Anwendungsmöglichkeiten und Funktionsweisen statistischer Wahrscheinlichkeiten im Betrieb! Dieses Buch vermittelt Ihnen die grundlegenden Verfahren der Wahrscheinlichkeitsrechnung als auch der Wahrscheinlichkeitsverteilungen und zeigt Ihnen deren praktische Anwendung in Betrieb und Ökonomie. Beispiele und Fragen mit Musterlösungen dienen dem weiteren Verständnis.Table of ContentsGrundlagen statistischer Wahrscheinlichkeiten in der Betriebswirtschaft: Grundbegriffe der Wahrscheinlichkeitsrechnung Diskrete Wahrscheinlichkeitsverteilungen - Binomial- und Hypergeometrische Verteilung Stetige Wahrscheinlichkeitsverteilung - Normalverteilung Intervallschätzung Notwendiger Stichprobenumfang Wahlforschung Hypothesentestverfahren Tabelle der Standardnormalverteilung Mathematische Grundlagen der induktiven Statistik Lösungen

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Book SynopsisIn den empirischen Sozialwissenschaften dienen die Methoden und Techniken der Statistik der Auswertung von Ergebnissen empirischer Untersuchungen und ermöglicht so die Beschreibung der quantitiativen Eigenschaften einer beobachteten und vollständig erfassten Gruppe. Das vorliegende Buch beschäftigt sich nahezu ausschließlich mit deskriptiver Statistik, welche sich mit der Aufbereitung und Beschreibung von Datenmengen und deren Verteilung befasst. Nach einer ausführlichen und grundlegenden Einführung in das Thema werden die wichtigsten Häufigkeitsverteilungen sowie die Maßzahlen zu deren Beschreibung dargestellt. Mit der linearen Einfachregression wird zuletzt der lineare funktionale Zusammenhang zweier Variabler erläutert. Zu jedem Kapitel gibt es Beispiele, Übungsaufgaben und eine Zusammenfassung. Im Anhang befinden sich mehrere Klausuren mit Lösungen, die in den letzten Jahren an der TU Dresden benutzt wurden.Table of Contents1 Grundlagen.- 1.1 Geisteswissenschaften und empirische Wissenschaften heute.- 1.2 Grundmethoden der empirischen Wissenschaften.- 1.2.1 Untersuchungsformen.- 1.2.2 Datenerhebungstechniken.- 1.2.3 Auswahlverfahren.- 1.3 Ablauf empirischer Sozialforschung: Der Forschungsprozess.- 1.3.1 Auswahl des Forschungsgegenstandes.- 1.3.2 Theoriebildung.- 1.3.3 Planung der Untersuchung.- 1.3.4 Durchführung der Untersuchung (Datenerhebung).- 1.3.5 Beschreibung und Zusammenfassung der Ergebnisse.- 1.3.6 Verallgemeinerung der Ergebnisse und Publikation.- 1.4 Einführung in die Forschungsstatistik.- 1.4.1 Statistische Gesetzmäßigkeiten.- 1.4.2 Grundlegende statistische Begriffe.- 1.4.3 Statistische Symbole.- 1.5 Begriff des Messens und der Messskalen.- 1.5.1 Der Begriff des Messens.- 1.5.2 Die Messniveaus.- 1.5.3 Die Bedeutung der Messniveaus für die Statistik.- 1.5.4 Gütekriterien der Messung.- Aufgaben.- 2 Empirische Häufigkeitsverteilungen.- 2.1 Häufigkeit und Verteilung.- 2.1.1 Das Aufstellen einer Häufigkeitstabelle.- 2.1.2 Absolute, relative und prozentuale Häufigkeiten.- 2.1.3 Die Häufigkeitsfunktion.- 2.1.4 Die Empirische Verteilungsfunktion.- 2.2 In Klassen eingeteilte Merkmale.- 2.2.1 Das Einteilen der Messwerte in Klassen.- 2.2.2 Aufstellen der Klassenhäufigkeiten.- 2.2.3 Offene Klassen.- 2.2.4 Exakte Klassengrenzen.- 2.2.5 Repräsentation einer Klasse durch die K1assenmitte.- 2.2.6 Informationsverlust durch Klasseneinteilung.- 2.3 Graphische Darstellungen von Häufigkeitsverteilungen.- 2.3.1 Das Stab-oder Balkendiagramm.- 2.3.2 Das Kreisdiagramm.- 2.3.3 Das Histogramm.- 2.3.4 Das Polygon.- 2.3.5 Typische Fonnen spezieller Verteilungen.- 2.4 Erkennen von Fehlinformation in statistischen Analysen.- 3 Maßzahlen eindimensionaler Verteilungen.- 3.1 Lageparameter.- 3.1.1 Das arithmetische Mittel.- 3.1.2 Der Median.- 3.1.3 Der Modus.- 3.1.4 Relative Positionen.- 3.1.5 Zulässige und optimale Lageparameter der einzelnen Messniveaus.- 3.2 Dispersionsparameter.- 3.2.1 Spannweite.- 3.2.2 Der (mittlere) Quartilabstand.- 3.2.3 Standardabweichung und Varianz.- 3.2.4 Der Variationskoeffizient zum Vergleich mehrerer Stichproben.- 3.2.5 Die Zusammenfassung von Varianzen.- 3.2.6 Gesamtvarianz, systematische Varianz und Fehlervarianz.- 3.2.7 Die Summe der quadratischen Abweichungen.- Aufgaben.- 4 Maßzahlen zweidimensionaler Verteilungen.- 4.1 Vorbemerkungen.- 4.1.1 Linearität.- 4.1.2 Die gemeinsame Verteilung.- 4.1.3 Ein einfaches Beispiel zur Darstellung bivariater Verteilungen.- 4.2 Korrelation.- 4.2.1 Intervallniveau.- 4.2.2 Ordinalniveau.- 4.3 Nomina1niveau.- 4.3.1 Tau (Goodman und Kruskal).- 4.3.2 Lambda.- 4.3.3 Kontingenzkoeffizient.- 4.3.4 Phi.- 4.3.5 Cramer’s V.- 4.4 Interpretation.- 5 Die lineare Einfachregression.- 5.1 Anpassen von Kurven.- 5.2 Vorhersage bei korrelierten Variablen.- 5.3 Methode der kleinsten Quadrate.- 5.3.1 Berechnung der Regressionsgeraden Gy/x.- 5.3.2 Berechnung der Regressionsgeraden Gx/y.- 5.4 Regressionskoeffizient, Korrelationskoeffizient und Varianz.- 5.5 Der Korrelationskoeffizient als Maß für die Güte der Regression.- 5.5.1 Die Varianz um die Regressionsgerade Sy/x2.- 5.5.2 Die Varianz auf der Regressionsgeraden $$ s_{\tilde y}^2 $$.- 5.6 Berechnung zweier Beispielaufgaben.- 5.6.1 Beispiel 1.- 5.6.2 Beispiel 2.- Weiterführende Literatur.- Anhang 1 Probeklausuren.- Anhang 2 Lösungen zu den Probeklausuren.

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Book SynopsisIsoperimetric, measure concentration and random process techniques appear at the basis of the modern understanding of Probability in Banach spaces. Based on these tools, the book presents a complete treatment of the main aspects of Probability in Banach spaces (integrability and limit theorems for vector valued random variables, boundedness and continuity of random processes) and of some of their links to Geometry of Banach spaces (via the type and cotype properties). Its purpose is to present some of the main aspects of this theory, from the foundations to the most important achievements. The main features of the investigation are the systematic use of isoperimetry and concentration of measure and abstract random process techniques (entropy and majorizing measures). Examples of these probabilistic tools and ideas to classical Banach space theory are further developed.Trade ReviewThis book gives an excellent, almost complete account of the whole subject of probability in Banach spaces, a branch of probability theory that has undergone vigorous development... There is no doubt in the reviewer's mind that this book [has] become a classic. MathSciNetAs the authors state, "this book tries to present some of the main aspects of the theory of probability in Banach spaces, from the foundation of the topic to the latest developments and current research questions''. The authors have succeeded admirably… This very comprehensive book develops a wide variety of the methods existing … in probability in Banach spaces. … It [has] become an event for mathematicians… Zentralblatt MATHTable of ContentsNotation.- 0. Isoperimetric Background and Generalities.- 1. Isoperimetric Inequalities and the Concentration of Measure Phenomenon.- 2. Generalities on Banach Space Valued Random Variables and Random Processes.- I. Banach Space Valued Random Variables and Their Strong Limiting Properties.- 3. Gaussian Random Variables.- 4. Rademacher Averages.- 5. Stable Random Variables.- 6 Sums of Independent Random Variables.- 7. The Strong Law of Large Numbers.- 8. The Law of the Iterated Logarithm.- II. Tightness of Vector Valued Random Variables and Regularity of Random Processes.- 9. Type and Cotype of Banach Spaces.- 10. The Central Limit Theorem.- 11. Regularity of Random Processes.- 12. Regularity of Gaussian and Stable Processes.- 13. Stationary Processes and Random Fourier Series.- 14. Empirical Process Methods in Probability in Banach Spaces.- 15. Applications to Banach Space Theory.- References.

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Book SynopsisIn Part I of this report the pointwise derivation of scalar set functions is investigated, first along the lines of R. DE POSSEL (abstract derivation basis) and A. P. MORSE (blankets); later certain concrete situations (e. g. , the interval basis) are studied. The principal tool is a Vitali property, whose precise form depends on the derivation property studied. The "halo" (defined at the beginning of Part I, Ch. IV) properties can serve to establish a Vitali property, or sometimes produce directly a derivation property. The main results established are the theorem of JESSEN-MARCINKIEWICZ-ZYGMUND (Part I, Ch. V) and the theorem of A. P. MORSE on the universal derivability of star blankets (Ch. VI) . . In Part II, points are at first discarded; the setting is somatic. It opens by treating an increasing stochastic basis with directed index sets (Th. I. 3) on which premartingales, semimartingales and martingales are defined. Convergence theorems, due largely to K. KRICKEBERG, are obtained using various types of convergence: stochastic, in the mean, in Lp-spaces, in ORLICZ spaces, and according to the order relation. We may mention in particular Th. II. 4. 7 on the stochastic convergence of a submartingale of bounded variation. To each theorem for martingales and semi-martingales there corresponds a theorem in the atomic case in the theory of cell (abstract interval) functions. The derivates concerned are global. Finally, in Ch.Table of ContentsI Pointwise Derivation.- I: Derivation Bases.- 1. Setting and general notation.- 2. dePossel’s derivation basis.- 3. Examples of bases.- 4. Pretopological notions.- 5. Comparison lemmas.- II: Derivation Theorems for ?-additive Set Functions under Assumptions of the Vitali Type.- 1. The individual Vitali assumption.- 2. The individual full derivation theorem for Radon or ?-fmite ?-integrals.- 3. The individual full derivation theorem for Radon measures.- 4. Class derivation theorems.- 5. Relation to Younovitch’s derivation theorem.- 6. The strong Vitali property.- 7. Half-regular and regular branches of a derivation basis.- III: The Converse Problem I: Covering Properties Deduced from Derivation Properties of ?-additive Set Functions.- 1. dePossel’s equivalence theorem.- 2. A necessary and sufficient condition for a weak derivation basis to derive a ?-finite ?-measure (Radon measure) ?.- 3. Younovitch’s equivalence theorem.- 4. A converse theorem for bases deriving the ?(q)-functions, q ? 1.- IV: Halo Assumptions in Derivation Theory. Converse Problem II.- 1. A. P. Morse’s halo properties.- 2. Abstract version of the strong Vitali theorem modelled after Banach.- 3. Abstract version of the strong Vitali theorem modelled after Carathéodory.- 4. Weak halo evanescence condition.- 5. Further criteria for the validity of the Density Theorem involving the weak halo.- 6. An individual derivability condition of Busemann-Feller type.- 7. The weak halo property in general bases.- 8. Product invariance of a weak halo property.- V: The Interval Basis. The Theorem of Jessen-Marcin-Kiewicz-Zygmund.- 1. The interval basis as a weak derivation basis.- 2. Theorem of Jessen-Marcinkiewicz-Zygmund.- 3. Properties of the halo function as consequences of derivation properties.- 4. Saks’ counterexample.- 5. The parallelepipedon basis.- 6. Saks’ “rarity” theorem.- VI: A. P. Morse’s Blankets.- 1. Nets.- 2. Hives.- 3. Fundamental covering theorems.- 4. Star blankets.- II Martingales and Cell Functions.- I: Theory without an Intervening Measure.- 1. Additive functions.- 2. ?-additive functions.- 3. Premartingales, semi-martingales, and martingales.- 4. Ordered space of martingales of basis(??).- 5. Integrals of premartingales.- 6. Martingales and additive functions.- 7. ?-additive martingales.- 8. Induced martingales.- 9. Premartingales and cell functions.- 10. Integrals of cell functions.- 11. Convergence theorems for martingales of bounded variation when ? is a measure algebra.- II: Theory in a Measure Space without Vitali Conditions.- 1. Preliminaries.- 2. Absolutely continuous and singular premartingales.- 3. Stochastic processes.- 4. Stochastic convergence.- 5. Mean convergence of order 1.- 6. Convergence in Orlicz spaces.- 7. Cell functions.- III: Theory in a Measure Space with Vitali Conditions.- 1. Preliminaries and definitions.- 2. Vitali conditions.- 3. Order convergence of martingales.- 4. Necessity of the Vitali conditions.- 5. Order convergence of submartingales.- 6. Order convergence of cell functions.- IV: Applications.- 1. Pointwise setting.- 2. Specifically pointwise concepts and results. Convergence almost everywhere.- 3. Martingales in the classical sense.- 4. Product spaces.- 5. The Radon- Nikodym integrand defined as a derivate.- 6. Representation of the spaces Lx as spaces of cell functions.- 7. Pointwise derivation of cell functions.- 8. Examples of concrete cell bases.- 9. Stochastic bases on a group.- Complements.- 1°. Derivation of vector-valued integrals.- 2°. Functional derivatives.- 3°. Topologies generated by measures.- 4°. Vitali’s theorem for invariant measures.- 5°. Global derivatives in locally compact topological groups..- 6°. Submartingales with decreasing stochastic bases.- 7°. Vector-valued martingales and derivation.- 9°. Derivation of measures.

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Book SynopsisOne SCI\'ice mathematics bas rendered the 'Et moi, ...si j'avait su comment en revcnir. je n'y serais point aile: human race. It bas put common sc:nsc back where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded n- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Hcavisidc Mathematics is a tool for thought. A highly necessary tool in a world where 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 statements as: 'One service topology has rendered mathematical physics .. :; 'One service logic has rendered com- 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.Table of Contents0. Preliminary Informationh.- 0.1 Probability Space.- 0.2 Random Functions and Processes.- 0.3 Conditional Probabilities.- 0.4 Independence.- 1. Sums of Independent Random Variables.- 1.1 Main Inequalities.- 1.2 Renewal Scheme.- 1.3 Random Walks. Recurrence.- 1.4 Distribution of Ladder Functions.- 2. General Processes with Independent Increments (Random Measures).- 2.1 Nonnegative Random Measures with Independent Values (r.m.i.v.).- 2.2 Random Measures with Alternating Signs.- 2.3 Stochastic Integrals and Countably Additive r.m.i.v.- 2.4 Random Linear Functional and Generalized Functions.- 3. Processes with Independent Increments. General Properties.- 3.1 Decomposition of a Process. Properties of Sample Functions.- 3.2 Stochastically Continuous Processes.- 3.3 Properties of Sample Functions.- 3.4 Locally Homogeneous Processes with Independent Increments.- 4. Homogeneous Processes.- 4.1 General Properties.- 4.2 Additive Functionals.- 4.3 Composed Poisson Process.- 4.4 Homogeneous Processes in R.- 5. Multiplicative Processes.- 5.1 Definition and General Properties.- 5.2 Multiplicative Processes in Abelian Groups.- 5.3 Stochastic Semigroups of Linear Operators in Rd.- Notes.- References.

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Book SynopsisThe principles of symmetry and self-symmetry are expressed in fractals, the subject of study in dimension theory. This book introduces an area of research which has recently appeared in the interface between dimension theory and the theory of dynamical systems, focusing on invariant fractals.

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a huge range and FREE tracked UK delivery on ALL orders.

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Book SynopsisThis book is an evolution from my book A First Course in Information Theory published in 2002 when network coding was still at its infancy.Trade ReviewFrom the reviews: "This book could serve as a reference in the general area of information theory and would be of interest to electrical engineers, computer engineers, or computer scientists with an interest in information theory. Each chapter has an appropriate problem set at the end and a brief paragraph that provides insight into the historical significance of the material covered therein. … Summing Up: Recommended. Upper-division undergraduate through professional collections." (J. Beidler, Choice, Vol. 46 (9), May, 2009) "The book consisting of 21 chapters is divided into two parts. Part I, Components of Information Theory … . Part II Fundamentals of Network Coding … . A comprehensive instructor’s manual is available. This is a well planned comprehensive book on the subject. The writing style of the author is quite reader friendly. … it is a welcome addition to the subject and will be very useful to students as well as to the researchers in the field." (Arjun K. Gupta, Zentralblatt MATH, Vol. 1154, 2009)Table of ContentsThe Science of Information.- The Science of Information.- Fundamentals of Network Coding.- Information Measures.- Information Measures.- Zero-Error Data Compression.- Weak Typicality.- Strong Typicality.- Discrete Memoryless Channels.- Rate-Distortion Theory.- The Blahut–Arimoto Algorithms.- Differential Entropy.- Continuous-Valued Channels.- Markov Structures.- Information Inequalities.- Shannon-Type Inequalities.- Beyond Shannon-Type Inequalities.- Entropy and Groups.- Fundamentals of Network Coding.- The Max-Flow Bound.- Single-Source Linear Network Coding: Acyclic Networks.- Single-Source Linear Network Coding: Cyclic Networks.- Multi-source Network Coding.

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a huge range and FREE tracked UK delivery on ALL orders.

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a huge range and FREE tracked UK delivery on ALL orders.

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Book SynopsisLinear Quadratic Differential Games is an assessment of the state of the art in its field and modern book on linear-quadratic game theory, one of the most commonly used tools for modelling and analysing strategic decision making problems in economics and management.Table of ContentsPreface. Notation and symbols. 1 Introduction. 1.1 Historical perspective. 1.2 How to use this book. 1.3 Outline of this book. 1.4 Notes and references. 2 Linear algebra. 2.1 Basic concepts in linear algebra. 2.2 Eigenvalues and eigenvectors. 2.3 Complex eigenvalues. 2.4 Cayley–Hamilton theorem. 2.5 Invariant subspaces and Jordan canonical form. 2.6 Semi-definite matrices. 2.7 Algebraic Riccati equations. 2.8 Notes and references. 2.9 Exercises. 2.10 Appendix. 3 Dynamical systems. 3.1 Description of linear dynamical systems. 3.2 Existence–uniqueness results for differential equations. 3.2.1 General case. 3.2.2 Control theoretic extensions. 3.3 Stability theory: general case. 3.4 Stability theory of planar systems. 3.5 Geometric concepts. 3.6 Performance specifications. 3.7 Examples of differential games. 3.8 Information, commitment and strategies. 3.9 Notes and references. 3.10 Exercises. 3.11 Appendix. 4 Optimization techniques. 4.1 Optimization of functions. 4.2 The Euler–Lagrange equation. 4.3 Pontryagin’s maximum principle. 4.4 Dynamic programming principle. 4.5 Solving optimal control problems. 4.6 Notes and references. 4.7 Exercises. 4.8 Appendix. 5 Regular linear quadratic optimal control. 5.1 Problem statement. 5.2 Finite-planning horizon. 5.3 Riccati differential equations. 5.4 Infinite-planning horizon. 5.5 Convergence results. 5.6 Notes and references. 5.7 Exercises. 5.8 Appendix. 6 Cooperative games. 6.1 Pareto solutions. 6.2 Bargaining concepts. 6.3 Nash bargaining solution. 6.4 Numerical solution. 6.5 Notes and references. 6.6 Exercises. 6.7 Appendix. 7 Non-cooperative open-loop information games. 7.1 Introduction. 7.2 Finite-planning horizon. 7.3 Open-loop Nash algebraic Riccati equations. 7.4 Infinite-planning horizon. 7.5 Computational aspects and illustrative examples. 7.6 Convergence results. 7.7 Scalar case. 7.8 Economics examples. 7.8.1 A simple government debt stabilization game. 7.8.2 A game on dynamic duopolistic competition. 7.9 Notes and references. 7.10 Exercises. 7.11 Appendix. 8 Non-cooperative feedback information games. 8.1 Introduction. 8.2 Finite-planning horizon. 8.3 Infinite-planning horizon. 8.4 Two-player scalar case. 8.5 Computational aspects. 8.5.1 Preliminaries. 8.5.2 A scalar numerical algorithm: the two-player case. 8.5.3 The N-player scalar case. 8.6 Convergence results for the two-player scalar case. 8.7 Notes and references. 8.8 Exercises. 8.9 Appendix. 9 Uncertain non-cooperative feedback information games. 9.1 Stochastic approach. 9.2 Deterministic approach: introduction. 9.3 The one-player case. 9.4 The one-player scalar case. 9.5 The two-player case. 9.6 A fishery management game. 9.7 A scalar numerical algorithm. 9.8 Stochastic interpretation. 9.9 Notes and references. 9.10 Exercises. 9.11 Appendix. References. Index.

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Book SynopsisModeling Random Processes for Engineers and Managers provides students with a gentle introduction to stochastic processes, emphasizing full explanations and many examples rather than formal mathematical theorems and proofs. The text offers an accessible entry into a very useful and versatile set of tools for dealing with uncertainty and variation. Many practical examples of models, as well as complete explanations of the thought process required to create them, motivate the presentation of the computational methods. In addition, the text contains a previously unpublished computational approach to solving many of the equations that occur in Markov processes. Modeling Random Processes is intended to serve as an introduction, but more advanced students can use the case studies and problems to expand their understanding of practical uses of the theory.Table of ContentsPreface ix 1 Probability Review 1 1.1 Interpreting and Using Probabilities 2 1.2 Sample Spaces and Events 3 1.3 Probability 4 1.4 Random Variables 6 1.5 Probability Distributions 6 1.6 Joint, Marginal, and Conditional Distributions 11 1.7 Expectation 14 1.8 Variance and Other Moments 16 1.9 The Law of Total Probability 18 1.10 Discrete Probability Distributions 20 1.11 Continuous Probability Distributions 23 1.12 Where Do Distributions Come From? 26 1.13 The Binomial Process 28 1.14 Recommended Reading 29 2 Formulating Markov Chain Models 32 2.1 An Example 33 2.2 Modeling the Progress of Time 34 2.3 Modeling Possibilities as States 36 2.4 Simplifying Assumptions 38 2.5 Modeling Changes of State as Transitions 40 2.6 Obtaining the Data 45 2.7 Another Example 46 2.8 A Case Study 47 2.9 Higher Order Markov Chains 50 2.10 Reducing the Number of States 52 2.11 Nonstationary Markov Chains 53 2.12 Other Example 54 2.13 Summary 67 2.14 Recommended Reading 67 3 Markov Chain Calculations 72 3.1 Walk Probabilities 73 3.2 Transition Probabilities 74 3.3 State Probabilities 78 3.4 A Numerical Example 79 3.5 Expected Number of Visits 80 3.6 Sojourn Times 82 3.7 First Passage and Return Probabilities 83 3.8 Computational Formulas for All Markov Chains 86 3.9 Special Classes of Markov Chains 86 3.10 Steady-State Probabilities 87 3.11 The Uses of Steady-State Results 92 3.12 Mean First Passage Times 93 3.13 Computational Formulas for Ergodic Markov Chains 96 3.14 Terminating Markov Chains 96 3.15 Expected Number of Visits 98 3.16 Expected Duration of a Terminating Process 99 3.17 Absorption Probabilities 100 3.18 Hit Probabilities 102 3.19 Conditional Mean First Passage Times to Absorbing States 103 3.20 Computational Formulas for Terminating Processes 105 3.21 Call Center Calculations 105 3.22 Classification Terminology 106 3.23 Additional Complications in Infinite Chains 111 3.24 Dealing with a Reducible Process 112 3.25 Periodic Chains 113 3.26 Ergodic Chains 114 3.27 Recommended Reading 115 4 Rewards on Markov Chains 119 4.1 Formulation 120 4.2 A Numerical Example 120 4.3 Expected Total Reward 121 4.4 Random Variable Rewards 124 4.5 Semi-Markov Processes 126 4.6 Limiting Results for Ergodic Processes 126 4.7 Total Reward for Terminating Processes 130 4.8 Case Study 132 4.9 Discounting 133 4.10 Case Study 135 4.11 Recommended Reading 137 5 Continuous Time Markov Processes 140 5.1 An Example 141 5.2 Interpreting Transition Rates 146 5.3 The Assumptions Reconsidered 149 5.4 Aging Does Not Affect the Transition Time 150 5.5 Competing Transitions 152 5.6 Sojourn Times 153 5.7 Embedded Markov Chains 154 5.8 Deriving the Differential Equations 155 5.9 Solving the Differential Equations 157 5.10 State Probabilities 159 5.11 First Passage Probability Functions 159 5.12 State Classification 160 5.13 Steady-State Probabilities 161 5.14 Other Computable Quantities 163 5.15 Case Study 165 5.16 Birth-Death Processes 167 5.17 The Poisson Process 169 5.18 Properties of Poisson Processes 171 5.19 Khintchine’s Theorem 172 5.20 Phase-Type Distributions 173 5.21 Conclusions 175 5.22 Recommended Reading 175 6 Queueing Models 179 6.1 An Example 180 6.2 General Characteristics 182 6.3 Performance Measures 186 6.4 Relations Among Performance Measures 188 6.5 Little’s Formula 190 6.6 Markovian Queueing Models 191 6.7 The M/M/1 Model 193 6.8 The Significance of Traffic Intensity 198 6.9 Unnormalized Solutions 200 6.10 Limited Queue Capacity 202 6.11 Multiple Servers 204 6.12 Is It Better to Merge or Separate Servers? 207 6.13 Which is Better: More Servers or Faster Servers 208 6.14 Case Study: A Grain Elevator 209 6.15 The M/M/c/c and M/M/1 Models 210 6.16 Finite Sources 212 6.17 The Machine Repairmen Model 214 6.18 Numerical Calculations Using a Spreadsheet 214 6.19 Queue Discipline Variations 217 6.20 Non-Markovian Queues 218 6.21 The M/G/1 Model 219 6.22 Approximate Solutions for Other Models 220 6.23 Conclusion 221 6.24 Recommended Reading 221 7 Networks of Queues 225 7.1 Open Networks of Markovian Queues 226 7.2 An Example Open Network 227 7.3 Extensions 228 7.4 Closed Networks 229 7.5 A Preliminary Example 229 7.6 Relative Arrival Rates 230 7.7 Unnormalized Solutions for Individual Stations 232 7.8 Assembling the Pieces of the Solution 234 7.9 Calculating the Normalization Constant 235 7.10 Performance Measures for Closed Networks 237 7.11 Creating a Closed Model 239 7.12 Case Study 242 7.13 Extensions 247 7.14 Approximate Methods 247 7.15 Recommended Reading 248 8 Using the Transition Diagram to Compute 251 8.1 An Example 252 8.2 Definitions 254 8.3 Steady-State Probabilities 258 8.4 How to Generate All In-trees 259 8.5 Check Your Understanding 262 8.6 Generalization to Other Quantities 263 8.7 Mean First Passage Times 264 8.8 Results for Terminating Processes 265 8.9 How to Simplify the Arithmetic 266 8.10 How to Systematically Generate r-Forests 267 8.11 Summary of Results 267 8.12 How to Remember the Formulas 268 8.13 Advanced Topics 268 8.14 Recommended Reading 269 Appendix 1 271 Appendix 2 278 Index 300

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Book SynopsisAn extensive update to a classic text Stochastic geometry and spatial statistics play a fundamental role in many modern branches of physics, materials sciences, engineering, biology and environmental sciences.Table of ContentsForeword to the first edition xiii From the preface to the first edition xvii Preface to the second edition xix Preface to the third edition xxi Notation xxiii 1 Mathematical foundations 1 1.1 Set theory 1 1.2 Topology in Euclidean spaces 3 1.3 Operations on subsets of Euclidean space 5 1.4 Mathematical morphology and image analysis 7 1.5 Euclidean isometries 9 1.6 Convex sets in Euclidean spaces 10 1.7 Functions describing convex sets 17 1.8 Polyconvex sets 24 1.9 Measure and integration theory 27 2 Point processes I: The Poisson point process 35 2.1 Introduction 35 2.2 The binomial point process 36 2.3 The homogeneous Poisson point process 41 2.4 The inhomogeneous and general Poisson point process 51 2.5 Simulation of Poisson point processes 53 2.6 Statistics for the homogeneous Poisson point process 55 3 Random closed sets I: The Boolean model 64 3.1 Introduction and basic properties 64 3.2 The Boolean model with convex grains 78 3.3 Coverage and connectivity 89 3.4 Statistics 95 3.5 Generalisations and variations 103 3.6 Hints for practical applications 106 4 Point processes II: General theory 108 4.1 Basic properties 108 4.2 Marked point processes 116 4.3 Moment measures and related quantities 120 4.4 Palm distributions 127 4.5 The second moment measure 139 4.6 Summary characteristics 143 4.7 Introduction to statistics for stationary spatial point processes 145 4.8 General point processes 156 5 Point processes III: Models 158 5.1 Operations on point processes 158 5.2 Doubly stochastic Poisson processes (Cox processes) 166 5.3 Neyman–Scott processes 171 5.4 Hard-core point processes 176 5.5 Gibbs point processes 178 5.6 Shot-noise fields 200 6 Random closed sets II: The general case 205 6.1 Basic properties 205 6.2 Random compact sets 213 6.3 Characteristics for stationary and isotropic random closed sets 216 6.4 Nonparametric statistics for stationary random closed sets 230 6.5 Germ–grain models 237 6.6 Other random closed set models 255 6.7 Stochastic reconstruction of random sets 276 7 Random measures 279 7.1 Fundamentals 279 7.2 Moment measures and related characteristics 284 7.3 Examples of random measures 286 8 Line, fibre and surface processes 297 8.1 Introduction 297 8.2 Flat processes 302 8.3 Planar fibre processes 314 8.4 Spatial fibre processes 330 8.5 Surface processes 333 8.6 Marked fibre and surface processes 339 9 Random tessellations, geometrical networks and graphs 343 9.1 Introduction and definitions 343 9.2 Mathematical models for random tessellations 346 9.3 General ideas and results for stationary planar tessellations 357 9.4 Mean-value formulae for stationary spatial tessellations 367 9.5 Poisson line and plane tessellations 370 9.6 STIT tessellations 375 9.7 Poisson-Voronoi and Delaunay tessellations 376 9.8 Laguerre tessellations 386 9.9 Johnson–Mehl tessellations 388 9.10 Statistics for stationary tessellations 390 9.11 Random geometrical networks 397 9.12 Random graphs 402 10 Stereology 411 10.1 Introduction 411 10.2 The fundamental mean-value formulae of stereology 413 10.3 Stereological mean-value formulae for germ–grain models 421 10.4 Stereological methods for spatial systems of balls 425 10.5 Stereological problems for nonspherical grains (shape-and-size problems) 436 10.6 Stereology for spatial tessellations 440 10.7 Second-order characteristics and directional distributions 444 References 453 Author index 507 Subject index 521

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