Stochastics Books

Book SynopsisThis book provides a comprehensive up-to-date presentation of some of the classical areas of reliability, based on a more advanced probabilistic framework using the modern theory of stochastic processes.Trade Review This is an excellent book on mathematical, statistical and stochastic models in reliability. The authors have done an excellent job of unifying some of the stochastic models in reliability. The book is a good reference book but may not be suitable as a textbook for students in professional fields such as engineering. This book may be used for graduate level seminar courses for students who have had at least the first course in stochastic processes and some knowledge of reliability mathematics. It should be a good reference book for researchers in reliability mathematics.--Mathematical ReviewsTable of ContentsIntroduction.- Basic Reliability Theory.- Stochastic Failure Models.- Availability Analysis of Complex Systems.- Maintenance Optimization.

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Book SynopsisThis text is concerned with Bayesian learning, inference and forecasting in dynamic environments.Table of Contentsto the DLM: The First-Order Polynomial Model.- to the DLM: The Dynamic Regression Model.- The Dynamic Linear Model.- Univariate Time Series DLM Theory.- Model Specification and Design.- Polynomial Trend Models.- Seasonal Models.- Regression, Autoregression, and Related Models.- Illustrations and Extensions of Standard DLMs.- Intervention and Monitoring.- Multi-Process Models.- Non-Linear Dynamic Models: Analytic and Numerical Approximations.- Exponential Family Dynamic Models.- Simulation-Based Methods in Dynamic Models.- Multivariate Modelling and Forecasting.- Distribution Theory and Linear Algebra.

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Book SynopsisThis book covers the basic results and methods in probability theory. This new edition offers updated content, 100 additional problems for solution, and a new chapter glimpsing further topics such as stable distributions, domains of attraction and martingales.Trade ReviewFrom the reviews of the second edition:"This is an excellent introductory book on random variables, with a wealth of examples and exercises. … The material is very well organized … . The text is remarkably well written, mathematically and aesthetically; layout and fonts make it a pleasant reading, and the examples are often enlightening. I think it will be a valuable support for students and instructors and it should definitely find a place in every good library." (Fabio Mainardi, The Mathematical Association of America, October, 2009)“…A worthwhile addition to the textbook pool, one that will guide the student safely through to a point of competence and ability to embark on a more advanced study…” (International Statistical Review, 2010, 78, 1, 134-159)“The book addresses a unique niche mathematically inclined students previously exposed to an introductory course in probability … . The writing style is lucid and easy to follow. … book is clearly directed toward mathematicians and the highly mathematically inclined scientist or engineer who might be induced to study the mathematics of probability or mathematical statistics. For those who find the classical mathematical pedagogy motivating or those requiring a comprehensive readable reference work on the mathematics of probability theory the book can be highly recommended.” (Thomas D. Sandry, Technometrics, Vol. 53 (1), February, 2011)“This book … is intended as an introductory graduate level textbook in probability for statistics majors. … This book provides an elaborate description and a collection of results in probability theory. … The level of this book is suitable for a graduate course. Overall all concepts are well discussed with full mathematical rigor. … has a good collection of most of the results related to probability theory, the price is very reasonable, and I will recommend this book to university mathematics and statistics libraries.” (Sounak Chakraborty, Journal of the American Statistical Association, Vol. 106 (495), September, 2011)Table of ContentsMultivariate Random Variables.- Conditioning.- Transforms.- Order Statistics.- The Multivariate Normal Distribution.- Convergence.- An Outlook on Further Topics.- The Poisson Process.

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Book SynopsisIntended as a self-contained introduction to measure theory, this textbook also includes a comprehensive treatment of integration on locally compact Hausdorff spaces, the analytic and Borel subsets of Polish spaces, and Haar measures on locally compact groups.Trade ReviewFrom the book reviews:“This textbook provides a comprehensive and consistent introduction to measure and integration theory. … The book can be recommended to anyone having basic knowledge of calculus and point-set topology. It is very self-contained, and can thus serve as an excellent reference book as well.” (Ville Suomala, Mathematical Reviews, July, 2014)“In this second edition, Cohn has updated his excellent introduction to measure theory … and has made this great textbook even better. Those readers unfamiliar with Cohn’s style will discover that his writing is lucid. … this is a wonderful text to learn measure theory from and I strongly recommend it.” (Tushar Das, MAA Reviews, June, 2014)Table of Contents1. Measures.- Algebras and sigma-algebras.- Measures.- Outer measures.- Lebesgue measure.- Completeness and regularity.- Dynkin classes.- 2. Functions and Integrals.- Measurable functions.- Properties that hold almost everywhere.- The integral.- Limit theorems.- The Riemann integral.- Measurable functions again, complex-valued functions, and image measures.- 3. Convergence.- Modes of Convergence.- Normed spaces.- Definition of L^p and L^p.- Properties of L^p and L-p.- Dual spaces.- 4. Signed and Complex Measures.- Signed and complex measures.- Absolute continuity.- Singularity.- Functions of bounded variation.- The duals of the L^p spaces.- 5. Product Measures.- Constructions.- Fubini’s theorem.- Applications.- 6. Differentiation.- Change of variable in R^d.- Differentiation of measures.- Differentiation of functions.- 7. Measures on Locally Compact Spaces.- Locally compact spaces.- The Riesz representation theorem.- Signed and complex measures; duality.- Additional properties of regular measures.- The µ^*-measurable sets and the dual of L^1.- Products of locally compact spaces.- 8. Polish Spaces and Analytic Sets.- Polish spaces.- Analytic sets.- The separation theorem and its consequences.- The measurability of analytic sets.- Cross sections.- Standard, analytic, Lusin, and Souslin spaces.- 9. Haar Measure.- Topological groups.- The existence and uniqueness of Haar measure.- The algebras L^1 (G) and M (G).- Appendices.- A. Notation and set theory.- B. Algebra.- C. Calculus and topology in R^d.- D. Topological spaces and metric spaces.- E. The Bochner integral.- F Liftings.- G The Banach-Tarski paradox.- H The Henstock-Kurzweil and McShane integralsBibliography.- Index of notation.- Index.

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Book SynopsisThe techniques and methods presented for knowledge elicitation, model construction and verification, modeling techniques and tricks, learning models from data, and analyses of models have all been developed and refined on the basis of numerous courses that the authors have held for practitioners worldwide.Trade ReviewFrom the book reviews:“The monograph concentrates on intelligent systems for decision support based on probabilistic models, including Bayesian networks and influence diagrams. … This monograph provides a review of recent state affairs of probabilistic networks that can be useful for professionals, practitioners, and researchers from diverse fields of statistics and related disciplines. I think it can be used as a textbook in its own right for an upper level undergraduate course, especially for a reading course.” (Technometrics, Vol. 55 (2), May, 2013)Table of ContentsIntroduction.- Networks.- Probabilities.- Probabilistic Networks.- Solving Probabilistic Networks.- Eliciting the Model.- Modeling Techniques.- Data-Driven Modeling.- Conflict Analysis.- Sensitivity Analysis.- Value of Information Analysis.- Quick Reference to Model Construction.- List of Examples.- List of Figures.- List of Tables.- List of Symbols.- References.- Index.

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Trade Review“This book is the result … of a fruitful and long collaboration between a mathematician and a cell biologist. Capturing the best of both worlds, the book provides not only the biology and mathematical background for this topic, but also offers numerous examples which render it accessible to (post-graduate) students and researchers … . this book can be treated as an excellent textbook for a wide audience varying from students to lecturers.” (Irina Ioana Mohorianu, zbMATH 1312.92004, 2015)"This book treats the theory of several important types of branching processes and demonstrates their usefulness by many interesting and important applications. … Mathematical theory and biological applications are nicely interwoven. This text will be useful both to mathematicians (including graduate students) interested in relevant applications of stochastic processes in biology, as well as to mathematically oriented biologists working on the above mentioned topics." (R. Bürger, Monatshefte für Mathematik, Vol. 143 (1), 2004)"This is a significant book on applications of branching processes in biology, and it is highly recommended for those readers who are interested in the application and development of stochastic models, particularly those with interests in cellular and molecular biology." (Charles J. Mode, Siam Review, Vol. 45 (2), 2003)"This is a book written jointly by a mathematician and a cell biologist, who have collaborated on research in branching processes for more than a decade. In their own words, their monograph is intended for ‘mathematicians and statisticians who have had an introduction to stochastic processes but have forgotten much of their college biology, and for biologists who wish to collaborate with mathematicians and statisticians.’ They have largely succeeded in achieving their goal. The book can be strongly recommended to all students of branching processes; all libraries should have a copy." —ZENTRALBLATT MATH Table of ContentsMotivating Examples and Other Preliminaries.- Biological Background.- The Galton-Watson Process.- The Age-Dependent Process: Markov Case.- The Bellman-Harris Process.- Multitype Processes.- Branching Processes with Infinitely Many Types.- Genealogies of Branching Processes and their Applications.- References.

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Book SynopsisCompletely revised and greatly expanded, the new edition of this text takes readers who have been exposed to only basic courses in analysis through the modern general theory of random processes and stochastic integrals as used by systems theorists, electronic engineers and, more recently, those working in quantitative and mathematical finance.Trade Review“As supplementary reading for a second course or as s comprehensive (!) resource for the general theory of processes aimed at Ph. D. students and scholars, this second edition will stay a valuable resource.” (René L. Schilling, Mathematical Reviews, October, 2016)“This is a fundamental book in modern stochastic calculus and its applications: rich contents, well structured material, comprehensive coverage of all significant results given with complete proofs and well illustrated by examples, carefully written text. Hence, there are more than enough reasons to strongly recommend the book to a wide audience. Among them, there are good and motivated graduate university students. … Also, the book is an excellent reference book.” (Jordan M. Stoyanov, zbMATH 1338.60001, 2016)Table of ContentsPart I: Measure Theoretic Probability.- Measure Integral.- Probabilities and Expectation.- Part II: Stochastic Processes.- Filtrations, Stopping Times and Stochastic Processes.- Martingales in Discrete Time.- Martingales in Continuous Time.- The Classification of Stopping Times.- The Progressive, Optional and Predicable -Algebras.- Part III: Stochastic Integration.- Processes of Finite Variation.- The Doob-Meyer Decomposition.- The Structure of Square Integrable Martingales.- Quadratic Variation and Semimartingales.- The Stochastic Integral.- Random Measures.- Part IV: Stochastic Differential Equations.- Ito's Differential Rule.- The Exponential Formula and Girsanov's Theorem.- Lipschitz Stochastic Differential Equations.- Markov Properties of SDEs.- Weak Solutions of SDEs.- Backward Stochastic Differential Equations.- Part V: Applications.- Control of a Single Jump.- Optimal Control of Drifts and Jump Rates.- Filtering. Part VI: Appendices.

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Book SynopsisThis two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions.Volume I of the book is entirely devoted to the theory of mean field games without a common noise. The first half of the volume provides a self-contained introduction to mean field games, starting from concrete illustrations of games with a finite number of players, and ending with ready-for-use solvability results. Readers are provided with the tools necessary for the solution of forward-backward stochastic differential equations of the McKean-Vlasov type at the core of the probabilistic approach. The second half of this volume focuses on the main principles of analysis on the Wasserstein space. It includes Lions' approach to the Wasserstein differential calculus, and the applications of its results to the analysis of stochastic mean field control problems. Together, both Volume I and Volume II will greatly benefit mathematical graduate students and researchers interested in mean field games. The authors provide a detailed road map through the book allowing different access points for different readers and building up the level of technical detail. The accessible approach and overview will allow interested researchers in the applied sciences to obtain a clear overview of the state of the art in mean field games.Trade Review“The text is very well-written and can be used to study the theory on various levels. It develops systematically from the wealth of motivating examples and heuristical considerations, through the carefully chosen collection of in-depth explained preliminaries, to the extensive nontrivial theory explained in full detail. … The book is highly recommended for those interested in the foundations and the up-to-date development of MFGs, as well as in the general area of stochastic control and related issues of analysis and probability.” (Vassili, Mathematical Reviews, January, 2019)Table of ContentsPreface to Volume I.- Part I: The Probabilistic Approach to Mean Field Games.- Learning by Examples: What is a Mean Field Game?.- Probabilistic Approach to Stochastic Differential Games.- Stochastic Differential Mean Field Games.- FBSDEs and the Solution of MFGs without Common Noise.- Part II: Analysis on Wasserstein Space and Mean Field Control.- Spaces of Measures and Related Differential Calculus.- Optimal Control of SDEs of McKean-Vlasov Type.- Epologue to Volume I.- Extensions for Volume I. References.- Indices.

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Book SynopsisFocussing on stochastic models for the spread of infectious diseases in a human population, this book is the outcome of a two-week ICPAM/CIMPA school on "Stochastic models of epidemics" which took place in Ziguinchor, Senegal, December 5–16, 2015. The text is divided into four parts, each based on one of the courses given at the school: homogeneous models (Tom Britton and Etienne Pardoux), two-level mixing models (David Sirl and Frank Ball), epidemics on graphs (Viet Chi Tran), and statistics for epidemic models (Catherine Larédo). The CIMPA school was aimed at PhD students and Post Docs in the mathematical sciences. Parts (or all) of this book can be used as the basis for traditional or individual reading courses on the topic. For this reason, examples and exercises (some with solutions) are provided throughout.Table of Contents- Part I Stochastic Epidemics in a Homogeneous Community. - Introduction. - Stochastic Epidemic Models. - Markov Models. - General Closed Models. - Open Markov Models. - Part II Stochastic SIR Epidemics in Structured Populations. - Introduction. - Single Population Epidemics. - The Households Model. - A General Two-Level Mixing Model. - Part III Stochastic Epidemics in a Heterogeneous Community. - Introduction. - Random Graphs. - The Reproduction Number R0. - SIR Epidemics on Configuration Model Graphs. - Statistical Description of Epidemics Spreading on Networks: The Case of Cuban HIV. - Part IV Statistical Inference for Epidemic Processes in a Homogeneous Community. - Introduction. - Observations and Asymptotic Frameworks. - Inference for Markov Chain Epidemic Models. - Inference Based on the Diffusion Approximation of Epidemic Models. - Inference for Continuous Time SIR models.

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Book SynopsisThis textbook provides a broad overview of the present state of insurance mathematics and some related topics in risk management, financial mathematics and probability. Both non-life and life aspects are covered. The emphasis is on probability and modeling rather than statistics and practical implementation. Aimed at the graduate level, pointing in part to current research topics, it can potentially replace other textbooks on basic non-life insurance mathematics and advanced risk management methods in non-life insurance. Based on chapters selected according to the particular topics in mind, the book may serve as a source for introductory courses to insurance mathematics for non-specialists, advanced courses for actuarial students, or courses on probabilistic aspects of risk. It will also be useful for practitioners and students/researchers in related areas such as finance and statistics who wish to get an overview of the general area of mathematical modeling and analysis in insurance.Table of ContentsBasics.- Experience Rating.- Sums and Aggregate Claims.- Ruin Theory.- Markov Models in Life Insurance.- Financial Mathematics in Life Insurance.- Special Studies in Life Insurance.- Orderings and Comparisons.- Extreme Value Theory.- Dependence and Further Topics in Risk Management.- Stochastic Control in Non-Life Insurance.- Stochastic Control in Life Insurance.- Selected Further Topics.

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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 Synopsis

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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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