Stochastics Books

Book SynopsisMatrix-analytic methods (MAM) were introduced by Professor Marcel Neuts and have been applied to a variety of stochastic models since. In order to provide a clear and deep understanding of MAM while showing their power, this book presents MAM concepts and explains the results using a number of worked-out examples.This book’s approach will inform and kindle the interest of researchers attracted to this fertile field. To allow readers to practice and gain experience in the algorithmic and computational procedures of MAM, Introduction to Matrix Analytic Methods in Queues 1 provides a number of computational exercises. It also incorporates simulation as another tool for studying complex stochastic models, especially when the state space of the underlying stochastic models under analytic study grows exponentially.The book’s detailed approach will make it more accessible for readers interested in learning about MAM in stochastic models.Table of ContentsList of Notations ix Preface xi Chapter 1 Introduction 1 1.1. Probability concepts 2 1.1.1. Random variables 5 1.1.2. Discrete probability functions 6 1.1.3. Probability generating function 7 1.1.4. Continuous probability functions 7 1.1.5. Laplace transform and Laplace-Stieltjes transform 9 1.1.6. Measures of a random variable 10 1.2. Renewal process 11 1.2.1. Renewal function 12 1.2.2. Terminating renewal process 15 1.2.3. Poisson process 16 1.3. Matrix analysis 18 1.3.1. Basics 18 1.3.2. Eigenvalues and eigenvectors 23 1.3.3. Partitioned matrices 27 1.3.4. Matrix differentiation 28 1.3.5. Exponential matrix 30 1.3.6. Kronecker products and Kronecker sums 32 1.3.7. Vectorization (or direct sums) of matrices 33 Chapter 2 Markov Chains 35 2.1. Discrete-time Markov chains (DTMC) 36 2.1.1. Basic concepts, key definitions and results 36 2.1.2. Computation of the steady-state probability vector of DTMC 43 2.1.3. Absorbing DTMC 45 2.1.4. Taboo probabilities in DTMC 47 2.2. Continuous-time Markov chain (CTMC) 48 2.2.1. Basic concepts, key definitions and results 48 2.2.2. Computation of exponential matrix 52 2.2.3. Computation of the limiting probabilities of CTMC 57 2.2.4. Computation of the mean first passage times 58 2.3. Semi-Markov and Markov renewal processes 61 Chapter 3 Discrete Phase Type Distributions 71 3.1. Discrete phase type (DPH) distribution 72 3.2. DPH renewal processes 92 3.3. Exercises 97 Chapter 4 Continuous Phase Type Distributions 101 4.1. Continuous phase type (CPH) distribution 101 4.2. CPH renewal process 120 4.3. Exercises 137 Chapter 5 Discrete-Batch Markovian Arrival Process 143 5.1. Discrete-batch Markovian arrival process (D-BMAP) 144 5.2. Counting process associated with the D-BMAP 152 5.3. Generation of D-MAP processes for numerical purposes 162 5.4. Exercises 165 Chapter 6 Continuous-Batch Markovian Arrival Process 171 6.1. Continuous-time batch Markovian arrival process (BMAP) 171 6.2. Counting processes associated with BMAP 177 6.3. Generation of MAP processes for numerical purposes 198 6.4. Exercises 206 Chapter 7 Matrix-Analytic Methods (Discrete-Time) 213 7.1. M/G/1-paradigm (scalar case) 215 7.2. M/G/1-paradigm (matrix case) 224 7.3. GI/M/1-paradigm (scalar case) 244 7.4. GI/M/1-paradigm (matrix case) 252 7.5. QBD process (scalar case) 268 7.6. QBD process (matrix case) 269 7.7. Exercises 278 Chapter 8. Matrix-Analytic Methods (Continuous-time) 291 8.1. M/G/1-type (scalar case) 291 8.2. M/G/1-type (matrix case) 295 8.3. GI/M/1-type (scalar case) 297 8.4. GI/M/1-type (matrix case) 300 8.5. QBD process (scalar case) 304 8.6. QBD process (matrix case) 305 8.7. Exercises 308 Chapter 9. Applications 321 9.1. Production and manufacturing 322 9.2. Service sectors 323 9.2.1. Healthcare 324 9.2.2. Artificial Intelligence and the Internet of Things 324 9.2.3. Biological and medicine 325 9.2.4. Telecommunications 325 9.2.5. Supply chain 325 9.2.6. Consumer issues 326 References 327 Index 335 Summary of Volume 2 339

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Book SynopsisThis book is devoted to the study of stochastic measures (SMs). An SM is a sigma-additive in probability random function, defined on a sigma-algebra of sets. SMs can be generated by the increments of random processes from many important classes such as square-integrable martingales and fractional Brownian motion, as well as alpha-stable processes. SMs include many well-known stochastic integrators as partial cases.General Stochastic Measures provides a comprehensive theoretical overview of SMs, including the basic properties of the integrals of real functions with respect to SMs. A number of results concerning the Besov regularity of SMs are presented, along with equations driven by SMs, types of solution approximation and the averaging principle. Integrals in the Hilbert space and symmetric integrals of random functions are also addressed.The results from this book are applicable to a wide range of stochastic processes, making it a useful reference text for researchers and postgraduate or postdoctoral students who specialize in stochastic analysis.Table of ContentsAbbreviations and Notations ix Introduction xi Chapter 1 Integration with Respect to Stochastic Measures 1 1.1. Preliminaries 1 1.2. Stochastic measures 2 1.2.1. Definition and examples of SMs 2 1.2.2. Convergence defined by an SM 5 1.3. Integration of deterministic functions 6 1.4. Limit theorems for integral of deterministic functions 11 1.4.1 Convergence of ∫A fn dμ 13 1.4.2 Convergence of ∫X fdμn 14 1.5. σ-finite stochastic measures 16 1.6. Riemann integral of a random function w.r.t. a deterministic measure 21 1.6.1. Definition of the integral 21 1.6.2. Interchange of the order of integration 27 1.6.3. Iterated integral and integration by parts 29 1.7. Exercises 32 1.8. Bibliographical notes 34 Chapter 2 Path Properties of Stochastic Measures 35 2.1. Sample functions of stochastic measures and Besov spaces 35 2.1.1. Besov spaces 35 2.1.2. Auxiliary lemmas 37 2.1.3 Stochastic measures on [0, 1] 42 2.1.4 Stochastic measures on [0, 1] d 44 2.2. Fourier series expansion of stochastic measures 46 2.2.1 Convergence of Fourier series of the process μ(t) 46 2.2.2. Convergence of stochastic integrals 49 2.3. Continuity of the integral 51 2.3.1. Estimate of an integral 51 2.3.2. Parameter dependent integral 54 2.3.3. Continuity with respect to the upper limit 55 2.4. Exercises 57 2.5. Bibliographical notes 59 Chapter 3 Equations Driven by Stochastic Measures 61 3.1 Parabolic equation in R (case dμσ (x)) 61 3.1.1. Problem and the main result 61 3.1.2 Lemma About the Hölder Continuity in X 70 3.1.3 Lemma about the Hölder continuity in t 75 3.2 Heat equation in Rd (case dμ(t)) 78 3.2.1. Additional estimate of an integral 78 3.2.2. Problem and the main result 80 3.2.3 Lemma About the Hölder Continuity in X 84 3.2.4 Lemma about the Hölder continuity in t 91 3.3 Wave equation in R (case dμ(x)) 99 3.3.1. Problem and the main result 99 3.3.2 Lemma About the Hölder Continuity in X 102 3.3.3 Lemma about the Hölder continuity in t 106 3.4 Wave equation in R (case dμ(t)) 108 3.4.1. Problem and the main result 108 3.4.2 Lemma About the Lipschiz Continuity in X 109 3.4.3 Lemma about the Hölder continuity in t 111 3.5 Parabolic evolution equation in R d (weak solution, case dμ(t)) 114 3.6. Exercises 119 3.7. Bibliographical notes 120 Chapter 4 Approximation of Solutions of the Equations 123 4.1 Parabolic equation in R (case dμ(x)) 123 4.1.1. Problem and the main result 123 4.1.2. Auxiliary lemmas 130 4.1.3. Examples 133 4.2 Heat equation in Rd (case dμ(t)) 135 4.2.1. Problem and the main result 135 4.2.2. Auxiliary lemma 138 4.2.3. Examples 139 4.3 Wave equation in R (case dμ(t)) 140 4.3.1. Approximation by using the convergence of paths of SMs 140 4.3.2. Approximation by using the Fourier partial sums 142 4.3.3. Approximation by using the Fejèr sums 149 4.3.4. Auxiliary lemma 151 4.3.5. Example 153 4.4. Exercises 154 4.5. Bibliographical notes 155 Chapter 5. Integration and Evolution Equations in Hilbert Spaces 157 5.1. Preliminaries 157 5.2. Equations and integral with a real-valued SM 160 5.2.1. Integral w.r.t. a real-valued SM 160 5.2.2. Evolution equations driven by a real-valued SM 163 5.3. Equations and integrals with a Hilbert space-valued SM 166 5.3.1. Integrals w.r.t. a U-valued SM 166 5.3.2. Evolution equations driven by a U-valued SM 171 5.4. Exercises 172 5.5. Bibliographical notes 173 Chapter 6 Symmetric Integrals 175 6.1. Introduction 175 6.2. SM has finite strong cubic variation 176 6.3. Stratonovich-type integral 177 6.4. SDE driven by an SM 181 6.5. Wong–Zakai approximation 183 6.6. Some counterexamples 188 6.7. Exercises 192 6.8. Bibliographical notes 193 Chapter 7 Averaging Principle 195 7.1. Heat equation 195 7.1.1. Introduction 195 7.1.2. The problem 196 7.1.3. Averaging principle 197 7.2. Equation with the symmetric integral 202 7.2.1. Introduction 202 7.2.2. Averaging principle 205 7.3. Exercises 211 7.4. Bibliographical notes 212 Chapter 8 Solutions to Exercises 213 References 231 Index 241

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Book SynopsisThis book illustrates a number of asymptotic and analytic approaches applied for the study of random evolutionary systems, and considers typical problems for specific examples. In this case, constructive mathematical models of natural processes are used, which more realistically describe the trajectories of diffusion-type processes, rather than those of the Wiener process. We examine models where particles have some free distance between two consecutive collisions. At the same time, we investigate two cases: the Markov evolutionary system, where the time during which the particle moves towards some direction is distributed exponentially with intensity parameter λ; and the semi-Markov evolutionary system, with arbitrary distribution of the switching process. Thus, the models investigated here describe the motion of particles with a finite speed and the proposed random evolutionary process with characteristics of a natural physical process: free run and finite propagation speed. In the proposed models, the number of possible directions of evolution can be finite or infinite.Table of ContentsPreface ix Introduction xi Chapter 1 Multidimensional Models of Kac Type 1 1.1. Definitions and basic properties 1 1.2. Moments of evolutionary process 8 1.3. Systems of Kolmogorov equations 17 1.4. Evolutionary operator and theorem about weak convergence to the measure of the Wiener process 23 Chapter 2 Symmetry of Markov Random Evolutionary Processes in Rn 29 2.1. Symmetrization: definition and properties 29 2.2. Examples of symmetric distributions in Rn and distributions on n + 1-hedra32 2.2.1. Symmetric distributions 32 2.2.2. Distributions on n + 1-hedra 35 Chapter 3 Hyperparabolic Equations, Integral Equation and Distribution for Markov Random Evolutionary Processes 39 3.1. Hyperparabolic equations and methods of solving Cauchy problems 39 3.2. Analytical solution of a hyperparabolic equation with real-analytic initial conditions 46 3.3. Integral representation of the hyperparabolic equation 57 3.4. Distribution function of evolutionary process 67 Chapter 4 Fading Markov Random Evolutionary Process 77 4.1. Definition of fading Markov random evolutionary process, its moments and limit distribution 77 4.2. Integral equation for a function from the fading random evolutionary process 89 4.3. Equations in partial derivatives for a function of the fading random evolutionary process 93 Chapter 5 Two Models of the Evolutionary Process 99 5.1. Evolution on a complex plane 99 5.2. Evolution with infinitely many directions 109 5.2.1. Symmetric case 110 5.2.2. Non-symmetric case 119 Chapter 6 Diffusion Process with Evolution and Its Parameter Estimation 125 6.1. Asymptotic diffusion environment 125 6.2. Approximation of a discrete Markov process in asymptotic diffusion environment 127 6.3. Parameter estimation of the limit process 132 Chapter 7 Filtration of Stationary Gaussian Statistical Experiments 135 7.1. Introduction 135 7.2. Stochastic difference equation of the process of filtration 137 7.3. Coefficient of filtration 138 7.4. Equation of optimal filtration 139 7.5. Characterization of a filtered signal 141 Chapter 8 Adapted Statistical Experiments with Random Change of Time 143 8.1. Introduction 143 8.2. Statistical experiments and evolutionary processes 144 8.3. Stochastic dynamics of statistical experiments 145 8.4. Adapted statistical experiments in series scheme 147 8.5. Convergence of the adapted statistical experiments 149 8.6. Scaling parameter estimation 154 8.7. Statistical estimations of the renewal intensity parameter 155 8.7.1. Poisson’s renewal process with parameter q =2 156 8.7.2. Stationary renewal process with delay, determined by the initial distribution function of the limit over jumps 156 8.7.3. Renewal processes with arbitrarily distributed renewal intervals 157 Chapter 9 Filtering of Stationary Gaussian Statistical Experiments 159 9.1. Stationary statistical experiments 159 9.2. Filtering of discrete Markov diffusion 161 9.3. The filtering error 164 9.4. The filtering empirical estimation 166 Chapter 10 Asymptotic Large Deviations for Markov Random Evolutionary Process 171 10.1. Asymptotic large deviations 171 10.2. Asymptotically stopped Markov random evolutionary process 191 10.3. Explicit representation for the normalizing function 206 Chapter 11 Asymptotic Large Deviations for Semi-Markov Random Evolutionary Processes 209 11.1. Recurrent semi-Markov random evolutionary processes 209 11.2. Asymptotic large deviations 212 Chapter 12 Heuristic Principles of Phase Merging in Reliability Analysis 221 12.1. The duplicated renewal system 221 12.2. The duplicated renewal system in the series scheme 222 12.3. Heuristic principles of the phase merging 223 12.4. The duplicated renewal system without failure 225 References 227 Index 233

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Book SynopsisThe aim of this book is to promote interaction between engineering, finance and insurance, as these three domains have many models and methods of solution in common for solving real-life problems. The authors point out the strict inter-relations that exist among the diffusion models used in engineering, finance and insurance. In each of the three fields, the basic diffusion models are presented and their strong similarities are discussed. Analytical, numerical and Monte Carlo simulation methods are explained with a view to applying them to obtain the solutions to the different problems presented in the book. Advanced topics such as nonlinear problems, Lévy processes and semi-Markov models in interactions with the diffusion models are discussed, as well as possible future interactions among engineering, finance and insurance. Contents 1. Diffusion Phenomena and Models.2. Probabilistic Models of Diffusion Processes.3. Solving Partial Differential Equations of Second Order.4. Problems in Finance.5. Basic PDE in Finance.6. Exotic and American Options Pricing Theory.7. Hitting Times for Diffusion Processes and Stochastic Models in Insurance.8. Numerical Methods.9. Advanced Topics in Engineering: Nonlinear Models.10. Lévy Processes.11. Advanced Topics in Insurance: Copula Models and VaR Techniques.12. Advanced Topics in Finance: Semi-Markov Models.13. Monte Carlo Semi-Markov Simulation Methods.Table of ContentsIntroduction xiii Chapter 1 Diffusion Phenomena and Models 1 1.1 General presentation of diffusion process 1 1.2 General balance equations 6 1.3 Heat conduction equation 10 1.4 Initial and boundary conditions 12 Chapter 2 Probabilistic Models of Diffusion Processes 17 2.1 Stochastic differentiation 17 2.2 Itô’s formula 19 2.3 Stochastic differential equations (SDE) 24 2.4 Itô and diffusion processes 28 2.5 Some particular cases of diffusion processes 32 2.6 Multidimensional diffusion processes 36 2.7 The Stroock–Varadhan martingale characterization of diffusions (Karlin and Taylor) 41 2.8 The Feynman–Kac formula (Platen and Heath) 42 Chapter 3 Solving Partial Differential Equations of Second Order 47 3.1 Basic definitions on PDE of second order 47 3.2 Solving the heat equation 51 3.3 Solution by the method of Laplace transform 65 3.4 Green’s functions 75 Chapter 4 Problems in Finance 85 4.1 Basic stochastic models for stock prices 85 4.2 The bond investments 90 4.3 Dynamic deterministic continuous time model for instantaneous interest rate 93 4.4 Stochastic continuous time dynamic model for instantaneous interest rate 98 4.5 Multidimensional Black and Scholes model 110 Chapter 5 Basic PDE in Finance 111 5.1 Introduction to option theory 111 5.2 Pricing the plain vanilla call with the Black–Scholes–Samuelson model 115 5.3 Pricing no plain vanilla calls with the Black-Scholes-Samuelson model 120 5.4 Zero-coupon pricing under the assumption of no arbitrage 127 Chapter 6 Exotic and American Options Pricing Theory 145 6.1 Introduction 145 6.2 The Garman–Kohlhagen formula 146 6.3 Binary or digital options 149 6.4 “Asset or nothing” options 150 6.5 Numerical examples 152 6.6 Path-dependent options 153 6.7 Multi-asset options 157 6.8 American options 165 Chapter 7 Hitting Times for Diffusion Processes and Stochastic Models in Insurance 177 7.1 Hitting or first passage times for some diffusion processes 177 7.2 Merton’s model for default risk 193 7.3 Risk diffusion models for insurance 201 Chapter 8 Numerical Methods 219 8.1 Introduction 219 8.2 Discretization and numerical differentiation 220 8.3 Finite difference methods 222 9.1 Nonlinear model in heat conduction 232 Chapter 9 Advanced Topics in Engineering: Nonlinear Models 231 9.2 Integral method applied to diffusive problems 233 9.3 Integral method applied to nonlinear problems 239 9.4 Use of transformations in nonlinear problems 243 Chapter 10 Lévy Processes 255 10.1 Motivation 255 10.2 Notion of characteristic functions 257 10.3 Lévy processes 257 10.4 Lévy–Khintchine formula 259 10.5 Examples of Lévy processes 261 10.6 Variance gamma (VG) process 264 10.7 The Brownian–Poisson model with jumps 266 10.8 Risk neutral measures for Lévy models in finance 275 10.9 Conclusion 276 Chapter 11 Advanced Topics in Insurance: Copula Models and VaR Techniques 277 11.1 Introduction 277 11.2 Sklar theorem (1959) 279 11.3 Particular cases and Fréchet bounds 280 11.4 Dependence 288 11.5 Applications in finance: pricing of the bivariate digital put option 293 11.6 VaR application in insurance 296 Chapter 12 Advanced Topics in Finance: Semi-Markov Models 307 12.1 Introduction 307 12.2 Homogeneous semi-Markov process 308 12.3 Semi-Markov option model 328 12.4 Semi-Markov VaR models 332 12.5 Conclusion 339 Chapter 13 Monte Carlo Semi-Markov Simulation Methods 341 13.1 Presentation of our simulation model 341 13.2 The semi-Markov Monte Carlo model in a homogeneous environment 345 13.3 A credit risk example 350 13.4 Semi-Markov Monte Carlo with initial recurrence backward time in homogeneous case 362 13.5 The SMMC applied to claim reserving problem 363 13.6 An example of claim reserving calculation 366 Conclusion 379 Bibliography 381 Index 393

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Book SynopsisNonlinear modelling has become increasingly important and widely used in economics. This valuable book brings together recent advances in the area including contributions covering cross-sectional studies of income distribution and discrete choice models, time series models of exchange rate dynamics and jump processes, and artificial neural network and genetic algorithm models of financial markets. Attention is given to the development of theoretical models as well as estimation and testing methods with a wide range of applications in micro and macroeconomics, labour and finance.The book provides valuable introductory material that is accessible to students and scholars interested in this exciting research area, as well as presenting the results of new and original research. Nonlinear Economic Models provides a sequel to Chaos and Nonlinear Models in Economics by the same editors.Trade Review'This collection provides valuable introductory material that is accessible to students and scholars interested in this research area.' -- Business HorizonsTable of ContentsContents: Part I: Introduction 1. Nonlinear Modelling: An Introduction Part II: Cross-sectional Applications 2. A Model of Income Distribution 3. Truncated Distribution Families 4. Betit: A Flexible Binary Choice Model 5. Estimation of Generalised Distributions 6. Age and the Distribution of Earnings 7. Count Data and Discrete Distributions Part III: Time Series Applications 8. A Model of the Real Exchange Rate 9. Jump Models and Higher Moments 10. A Topological Test of Chaos 11. Genetic Algorithms and Trading Rules Part IV: Neural Network Applications 12. Artificial Neural Networks 13. An ANN Model of the Stock Market 14. Exchange Rate Forecasting Models Index

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Book SynopsisThis textbook, now in its second revised and extended edition, presents the fundamental ideas and results of both probability theory and statistics. It comprises the material of a one-year course, which is addressed to students of mathematics and to scientists with an interest in the mathematical side of stochastics. The stochastic concepts, models and methods are motivated by examples and then developed and analysed systematically. Some measure theory is included, but this is done at an elementary level that is in accordance with the introductory character of the book. A large number of problems, now in part with solutions, offer applications and supplements to the text.Trade Review"The textbook is based on a series of lectures taught by the author for many years at the Mathematical Institute of the University of Munich. The material of the book covers two one-semester courses in probability and mathematical statistics, respectively. All chapters are equipped with exercises of varying degrees of difficulty that help to clarify the concepts.The first part of the book is an introduction to probability theory. The material is presented using little of the measure-theoretical background but rather application-oriented examples that preserve its introductory character. Topics range from classical probability distributions to conditional distributions and limit theorems. A short introduction to Markov chains is also given.The second part of the book gives an introduction to mathematical statistics and describes main statistical procedures: parameter and interval estimation, hypothesis testing, linear regression and basics of the analysis of variance approach.The book can be used by undergraduate mathematics majors but also by science and engineering students who wish not only to apply probability and statistics but also to understand how the methods work."Vladimir P. Kurenok, MathSciNet

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Book SynopsisThe book gives a systematical presentation of stochastic approximation methods for models of American-type options with general pay-off functions for discrete time Markov price processes. Advanced methods combining backward recurrence algorithms for computing of option rewards and general results on convergence of stochastic space skeleton and tree approximations for option rewards are applied to a variety of models of multivariate modulated Markov price processes. The principal novelty of presented results is based on consideration of multivariate modulated Markov price processes and general pay-off functions, which can depend not only on price but also an additional stochastic modulating index component, and use of minimal conditions of smoothness for transition probabilities and pay-off functions, compactness conditions for log-price processes and rate of growth conditions for pay-off functions. The book also contains an extended bibliography of works in the area. This book is the first volume of the comprehensive two volumes monograph. The second volume will present results on structural studies of optimal stopping domains, Monte Carlo based approximation reward algorithms, and convergence of American-type options for autoregressive and continuous time models, as well as results of the corresponding experimental studies.

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Book SynopsisThis book contains all of Wolfgang Doeblin's publications. In addition, it includes a reproduction of the pli cacheté on l'équation de Kolmogoroff and previously unpublished material that Doeblin wrote in 1940. The articles are accompanied by commentaries written by specialists in Doeblin's various areas of interest. The modern theory of probability developed between the two World Wars thanks to the very remarkable work of Kolmogorov, Khinchin, S.N. Bernstein, Romanovsky, von Mises, Hostinsky, Onicescu, Fréchet, Lévy and others, among whom one name shines particularly brightly, that of Wolfgang Doeblin (1915–1940). The work of this young mathematician, whose life was tragically cut short by the war, remains even now, and indeed will remain into the future, an exemplar of originality and of mathematical power. This book was conceived and in essence brought to fruition by Marc Yor before his death in 2014. It is dedicated to him.Trade Review“It attracted interest not only by mathematicians or some probability theory specialists, but also by the public, which normally would not be that interested in mathematics. … the editors gathered not only all published articles of W. Doeblin, but also some more papers, especially the ‘cahier jaune’ (yellow notebook). In comments and annotations the contents of the papers are put into today’s context by probability theory specialists. The book is excellent, a large and substantial work.” (Silke Göbel, zbMATH 1460.01019, 2021)Table of ContentsJ.-M. Bismut, Avant-propos.- H. Föllmer, Geleitwort.- B. Bru, Preface.- Part I The Life of Wolfgang Doeblin March 17, 1915 – June 21, 1940: B. Bru, Biographical Summary.- W. Doeblin, Autobiographical Note.- W. Doeblin (with remarks by B. Bru and E. Seneta), Notice sur les travaux.- List of Works of Wolfgang Doeblin.- Conversion Table.- T. Lindvall, W. Doeblin 1915–1940.- B. Bru and M. Yor, La vie de Doeblin et le pli cacheté 11.668.- B. Bru and M. Yor, Comments on the Life and Mathematical Legacy of Wolfgang Doeblin.-B. Bru and E. Seneta, Two Letters of W. Doeblin to A. N. Kolmogorov.- Part II Markov Chains: E. Seneta, Doeblin on Discrete Markov Chains.- M. Iosifescu, On Doeblin and Fortet’s paper “Sur des chaînes à liaisons complètes”.- M. Iosifescu, On Doeblin’s paper “Remarques sur la théorie métrique des fractions continues”.- E. Nummelin, Doeblin’s Theory of Markov Chains on a General Measurable State Space.- W. Doeblin, Sur les chaînes discrètes de Markoff [1936b].- W. Doeblin, Errata 1: Sur les chaînes discrètes de Markoff [1936c].- W. Doeblin, Errata 2: Sur les chaînes discrètes de Markoff [1936d].- W. Doeblin, Sur les chaînes de Markoff [1936e].- W. Doeblin and R. Fortet, Sur deux notes de MM. Kryloff et Bogoliouboff [1937e].- W. Doeblin, Éléments d’une théorie générale des chaînes constantes simples de Markoff [1937f].- W. Doeblin, Le cas discontinu des probabilités en chaîne [1937a].- W. Doeblin, Sur le cas continu des probabilités en chaîne [1937b].- W. Doeblin, and Robert Fortet Sur des chaînes à liaisons complètes [1937c].- W. Doeblin, Sur l’équation de Smoluchowsky [1937d].- W. Doeblin, Sur les propriétés asymptotiques de mouvements régis par certains types de chaînes simples [1938g] (reprint of [1937g] and [1937h]).- W. Doeblin, Sur l’équation matricielle A(t+s) = [A(t)A(s)] et ses applications aux probabilités en chaîne [1938a].-W. Doeblin, Sur l’équation matricielle A(t+s) = [A(t)A(s)] et ses applications au calcul des probabilités [1940a].- W. Doeblin, Sur deux problèmes de M. Kolmogoroff concernant les chaînes dénombrables [1938f].- W. Doeblin, Exposé de la théorie des chaînes simples constantes de Markoff à un nombre fini d’états [1938e].- W. Doeblin, Remarques sur la théorie métrique des fractions continues [1940d.- W. Doeblin, Éléments d’une théorie générale des chaînes simples constantes de Markoff [1940e].- Part III Sums of Independent Random Variables: D. M. Mason, Notes on Wolfgang Doeblin’s 1940 paper: L’ensemble de puissances d’une loi de probabilité.- W. Doeblin and P. Lévy, Sur les sommes de variables aléatoires indépendantes à dispersions bornées inférieurement [1936a].- W. Doeblin, Premiers éléments d’une étude systématique de l’ensemble de puissances d’une loi de probabilité [1938b].- W. Doeblin, Étude de l’ensemble de puissances d’une loi de probabilité [1938c].- W. Doeblin, Sur les sommes d’un grand nombre de vecteurs aléatoires [1938d].- W. Doeblin, Sur un problème de calcul des probabilités [1939b].- W. Doeblin, Sur les sommes d’un grand nombre de variables aléatoires indépendantes [1939c].- W. Doeblin, Sur l’ensemble de puissances d’une loi de probabilité (1940) [1940f].- W. Doeblin (with remarks by B. Bru and E. Seneta), Sur l’ensemble de puissances d’une loi de probabilité [1946].- Part IV Chapman’s Equations. Part IV.1 Documents Published Before the pli cacheté: W. Doeblin (with a remark by B. Bru), Sur certaines mouvements aléatoires discontinus [1939d].- W. Doeblin, Sur l’équation de Kolmogoroff [1938h].- W. Doeblin, Sur certains mouvements aléatoires [1939a].- W. Doeblin, Sur l’équation de Kolmogoroff [1940b].- W. Doeblin, Sur des mouvements mixtes [1940c].- Part IV.2 The pli cacheté: M. Yor, Présentation du pli cacheté.- W. Doeblin (with remarks by Bernard Bru and Marc Yor), Sur l’équation de Kolmogoroff, pli cacheté déposé le 26 février 1940, ouvert le 18 mai 2000 [1940g/2000].- B. Bru, Notes de lecture du pli cacheté.- W. Doeblin (with remarks by Bernard Bru), Exposé(s) sur l’équation de Chapman [1938i/2000].- Part IV.3 Archival Documents Related to the pli cacheté: B. Bru, Introduction to Doeblin’s “Sur la solution de M. Hostinský de l’équation de Chapman”.- W. Doeblin, Sur la solution de M. Hostinský de l’équation de Chapman [1940h/1993].- W.Doeblin (with an introduction by B. Bru and E. Seneta), Le cahier jaune: Recherche sur l’équation de Chapman. Propriétés communes aux mouvements régis par l’équation de Chapman [1940i/2020].- Part V Appendix: B. Bru and E. Seneta, Postface.- Acknowledgements.- Credits.- Bibliography.- Supplementary Reading.

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Book SynopsisThis richly illustrated book is an exploration of how chance and risk, on the one hand, and meaning or significance on the other, compete for the limelight in art, in philosophy, and in science. In modern society, prudence and probability calculation permeate our daily lives. Yet it is clear for all to see that neither cautious bank regulations nor mathematics have prevented economic crises from occurring time and again. Nicolas Bouleau argues that it is the meaning we assign to an event that determines the perceived risk, and that we generally turn a blind eye to this important fact, because the word "meaning" is itself awkward to explain. He tackles this fundamental question through examples taken from cultural fields ranging from painting, architecture, and music, to poetry, biology, and astronomy. This enables the reader to view overwhelming risks in a different light. Bouleau clarifies that the most important thing in a time of uncertainty is to think of prudence on a higher level, one that truly addresses the various subjective interpretations of the world.Trade ReviewFrom the reviews:“Risk and Meaning is a quirky book, both in its topic and its physical layout and the writing style that the author adopts … . it is a book that is mathematics-adjacent, and I think that anyone who is able to get past (or even relish) some of the quirkiness in the book will certainly find themselves with plenty to think about. And I’m not sure what more one could ask for in a book.” (Darren Glass, The Mathematical Association of America, November, 2011)Table of ContentsEntrance: Interpretation and Paradigms.- I. Cicero and Divination.- II. Cournot’s "Philosophic Probabilities".- III. Mathematical Probabilities.- IV. Democracy by Chance.- V. Gestalt, Structure, Pattern.- VI. The Third Dimension of Risk.- VII. ''Modern" Architecture.- VIII. The Ideal City.- IX. Daring the Abstract in Art.- X. Saussure or the Dread of Mathematical Probabilities.- XI. Jacques Monod’s Roulette.- XII. From Fortuitism to Animism.- XIII. The Slip as Fortuity and Meaning.- XIV. Guessing Astronomy.- XV. The Legitimacy of Science and Love.- Hints and Index.

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Book SynopsisFrom the Preface: Srinivasa Varadhan began his research career at the Indian Statistical Institute (ISI), Calcutta, where he started as a graduate student in 1959. His first paper appeared in Sankhyá, the Indian Journal of Statistics in 1962. Together with his fellow students V. S. Varadarajan, R. Ranga Rao and K. R. Parthasarathy, Varadhan began the study of probability on topological groups and on Hilbert spaces, and quickly gained an international reputation. At this time Varadhan realised that there are strong connections between Markov processes and differential equations, and in 1963 he came to the Courant Institute in New York, where he has stayed ever since. Here he began working with the probabilists Monroe Donsker and Marc Kac, and a graduate student named Daniel Stroock. He wrote a series of papers on the Martingale Problem and Diffusions together with Stroock, and another series of papers on Large Deviations together with Donsker. With this work Varadhan's reputation as one of the leading mathematicians of the time was firmly established. Since then he has contributed to several other areas of probability, analysis and physics, and collaborated with numerous distinguished mathematicians. Varadhan was awarded the Abel Prize in 2007. These Collected Works contain all his research papers over the half-century spanning 1962 to early 2012. Volume I includes the introductory material, the papers on limit theorems and review articles.Table of ContentsAutobiography: S. R. S. Varadhan.- Introduction: S. R. S. Varadhan.- Prize Citations.- Diffusion Theory by Daniel W. Stroock. - Large Deviations by Daniel W. Stroock.- Large Deviation and Homogenization by Fraydoun Rezakhanlou.- Varadhan's Work on Hydrodynamical Limits by Jeremy Quastel and Horng-Tzer Yau.- Book Review: Multidimensional Diffusion Processes by D. W. Stroock and S. R. S. Varadhan.- Limit Theorems: Limit theorems for sums of independent random variables with values in a Hilbert space.- On the category of indecomposable distributions on topological groups.- Probability distributions on locally compact abelian groups.- Extension of stationary stochastic processes.- Limit theorems in probability.- A limit theorem with strong mixing in Banach space and two applications to stochastic differential equations.- Limit theorems for random walks on Lie groups.- Martingale approach to some limit theorems.- Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions.- Bounding functions of Markov processes and the shortest queue problem.- Finite approximations to quantum systems.- Self-diffusion of a tagged particle in equilibrium for asymmetric mean zero random walk with simple exclusion.- Diffusive limit of a tagged particle in asymmetric simple exclusion processes.- A martingale proof of Dobrushin's theorem for non-homogeneous Markov chains.- Review Articles.- Diffusion processes, Stochastic processes: theory and methods.- Stochastic analysis and applications.- Large deviations and entropy, Entropy.- The role of weak convergence in probability theory.

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Book SynopsisFrom the Preface: Srinivasa Varadhan began his research career at the Indian Statistical Institute (ISI), Calcutta, where he started as a graduate student in 1959. His first paper appeared in Sankhyá, the Indian Journal of Statistics in 1962. Together with his fellow students V. S. Varadarajan, R. Ranga Rao and K. R. Parthasarathy, Varadhan began the study of probability on topological groups and on Hilbert spaces, and quickly gained an international reputation. At this time Varadhan realised that there are strong connections between Markov processes and differential equations, and in 1963 he came to the Courant Institute in New York, where he has stayed ever since. Here he began working with the probabilists Monroe Donsker and Marc Kac, and a graduate student named Daniel Stroock. He wrote a series of papers on the Martingale Problem and Diffusions together with Stroock, and another series of papers on Large Deviations together with Donsker. With this work Varadhan's reputation as one of the leading mathematicians of the time was firmly established. Since then he has contributed to several other areas of probability, analysis and physics, and collaborated with numerous distinguished mathematicians. Varadhan was awarded the Abel Prize in 2007. These Collected Works contain all his research papers over the half-century spanning 1962 to early 2012. Volume II includes the papers on PDE, SDE, diffusions, and random media.​​Table of ContentsVol. II: Diffusion processes with continuous coefficients - I (with D. W. Stroock).- Diffusion processes with continuous coefficients - II (with D. W. Stroock).- Diffusion processes with boundary conditions (with D. W. Stroock).- On degenerate elliptic-parabolic operators of second order and their associated diffusions (with D. W. Stroock).- On the support of diffusion processes with applications to the strong maximum principle (with D. W. Stroock).- Diffusion processes (with D. W. Stroock).- A probabilistic approach to Hp(Rd) (with D. W. Stroock).- Kac functional and Schrodinger equation (with K. L. Chung).- Brownian motion in a wedge with oblique reection (with R. J. Williams).- A multidimensional process involving local time (with A.S. Sznitman).- Etat fondamental et principe du maximum pour les operateurs elliptiques du second ordre dans des domaines generaux. [The ground state and maximum principle for second-order elliptic operators in general domains] (with H. Berestycki and L. Nirenberg).- The principal eigenvalue and maximum principle for second-order elliptic operators in general domains (with H. Berestycki and L. Nirenberg).- Diffusion semigroups and di_usion processes corresponding to degenerate divergence form operators (with J. Quastel).- Random Media.- Diffusion in regions with many small holes (with G. Papanicolaou).- Boundary value problems with rapidly oscillating random coefficients (with G. Papanicolaou).- Diffusions with random coefficients (with G. Papanicolaou).- Ohrnstein-Uhlenbeck process in a random potential (with G. Papanicolaou).- Large deviations for random walks in a random environment.- Random walks in a random environment.- Stochastic homogenization of Hamilton-Jacobi-Bellman equations (with E. Kosygina and F. Rezakhanlou).- Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium (with E. Kosygina).- Behavior of the solution of a random semilinear heat equation (with N. Zygouras).​

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Book SynopsisFrom the Preface: Srinivasa Varadhan began his research career at the Indian Statistical Institute (ISI), Calcutta, where he started as a graduate student in 1959. His first paper appeared in Sankhyá, the Indian Journal of Statistics in 1962. Together with his fellow students V. S. Varadarajan, R. Ranga Rao and K. R. Parthasarathy, Varadhan began the study of probability on topological groups and on Hilbert spaces, and quickly gained an international reputation. At this time Varadhan realised that there are strong connections between Markov processes and differential equations, and in 1963 he came to the Courant Institute in New York, where he has stayed ever since. Here he began working with the probabilists Monroe Donsker and Marc Kac, and a graduate student named Daniel Stroock. He wrote a series of papers on the Martingale Problem and Diffusions together with Stroock, and another series of papers on Large Deviations together with Donsker. With this work Varadhan's reputation as one of the leading mathematicians of the time was firmly established. Since then he has contributed to several other areas of probability, analysis and physics, and collaborated with numerous distinguished mathematicians. Varadhan was awarded the Abel Prize in 2007. These Collected Works contain all his research papers over the half-century spanning 1962 to early 2012.Volume III includes the papers on large deviations. ​​Table of ContentsLarge Deviations.- Asymptotic probabilities and differential equations.- On the behavior of the fundamental solution of the heat equation with variable coefficients .- Diffusion processes in a small time interval .- On a variational formula for the principal eigenvalue for operators with maximum principle.- Asymptotic evaluation of certain Markov process expectations for large time I.- Asymptotic evaluation of certain Markov process expectations for large time II.- Asymptotic evaluation of certain Wiener integrals for large time.- Asymptotics for the Wiener sausage.- Erratum: Asymptotics for the Wiener sausage.- Asymptotic evaluation of certain Markov process expectations for large time III.- On the principal eigenvalue of second-order elliptic differential operators.- On laws of the iterated logarithm for local times.- Some problems of large deviations.- On the number of distinct sites visited by a random walk.- A law of the iterated logarithm for total occupation times of transient Brownian motion.- Some problems of large deviations .- The polaron problem and large deviations.- Asymptotic evaluation of certain Markov process expectations for large time IV.- Asymptotics for the polaron.- Large deviations for stationary Gaussian processes.- Large deviations and applications.- Large deviations for non-interacting infinite-particle systems.- Some familiar examples for which the large deviation principle does not hold.- The large deviation principle for the Erdös-Rényi random graph.- Large deviations for random matrices. ​

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Book SynopsisFrom the Preface: Srinivasa Varadhan began his research career at the Indian Statistical Institute (ISI), Calcutta, where he started as a graduate student in 1959. His first paper appeared in Sankhyá, the Indian Journal of Statistics in 1962. Together with his fellow students V. S. Varadarajan, R. Ranga Rao and K. R. Parthasarathy, Varadhan began the study of probability on topological groups and on Hilbert spaces, and quickly gained an international reputation. At this time Varadhan realised that there are strong connections between Markov processes and differential equations, and in 1963 he came to the Courant Institute in New York, where he has stayed ever since. Here he began working with the probabilists Monroe Donsker and Marc Kac, and a graduate student named Daniel Stroock. He wrote a series of papers on the Martingale Problem and Diffusions together with Stroock, and another series of papers on Large Deviations together with Donsker. With this work Varadhan's reputation as one of the leading mathematicians of the time was firmly established. Since then he has contributed to several other areas of probability, analysis and physics, and collaborated with numerous distinguished mathematicians. Varadhan was awarded the Abel Prize in 2007. These Collected Works contain all his research papers over the half-century spanning 1962 to early 2012. Volume IV includes the papers on particle systems.Table of ContentsVolume 4: Particle Systems and Their Large Deviations.- Nonlinear diffusion limit for a system with nearest neighbor interaction.- Hydrodynamics and large deviation for simple exclusion processes.- Large deviations from a hydrodynamic scaling limit.- On the derivation of conservation laws for stochastic dynamics.- Scaling limits for interacting diffusions.- Scaling limit for interacting Ornstein-Uhlenbeck processes.- Entropy methods in hydrodynamical scaling.- Hydrodynamical limit for a Hamiltonian system with weak noise.- Nonlinear diffusion limit for a system with nearest neighbor interactions II.- Regularity of self-diffusion coefficient.- Entropy methods in hydrodynamic scaling.- Spectral gap for zero-range dynamics.- The complex story of simple exclusion.- Non-gradient models in hydrodynamic scaling.- Relative entropy and mixing properties of interacting particle systems.- Diffusive limit of lattice gas with mixing conditions.- Large deviations for the symmetric simple exclusion process in dimensions d > 3.- Large deviations for interacting particle systems.- Infinite particle systems and their scaling limits.- Lectures on hydrodynamic scaling.- Scaling limits of large interacting systems .- Asymptotic behavior of a tagged particle in simple exclusion processes.- Large deviation and hydrodynamic scaling.- Symmetric simple exclusion process: regularity of the self-diffusion coefficient.- Finite-dimensional approximation of the self-diffusion coefficient for the exclusion process.- Large deviations for the asymmetric simple exclusion process.- Diffusive behaviour of the equilibrium fluctuations in the asymmetric exclusion processes.- On viscosity and fluctuation-dissipation in exclusion processes.- Large deviations for the current and tagged particle in 1d nearest neighbor.- Symmetric simple exclusion.- List of Publications of S.R.S. Varadhan.- Acknowledgements. ​

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Book Synopsis01/07 This title is now available from Walter de Gruyter. Please see www.degruyter.com for more information. This book is devoted to stochastic operators in Hilbert space. A number of models in modern probability theory apply the notion of a stochastic operator in explicit or latent form. In this book, objects from the Gaussian case are considered. Therefore, it is useful to consider all random variables and elements as functionals from the Wiener process or its formal derivative, i.e. white noise. The book consists of five chapters. The first chapter is devoted to stochastic calculus and its main goal is to prepare the tools for solving stochastic equations. In the second chapter the structure of stochastic equations, mainly the structure of Gaussian strong linear operators, is studied. In chapter 3 the definition of the action of the stochastic operator on random elements in considered. Chapter 4 deals with the mathematical models in which the notions of stochastic calculus arise and in the final chapter the equation with random operators is considered.Table of ContentsFrontmatter -- Contents -- Introduction -- Chapter 1. Stochastic calculus -- Chapter 2. Random maps in Hilbert space -- Chapter 3. The composition of random maps -- Chapter 4. Stochastic analysis and quantum mechanics -- Chapter 5. Equations with random operators -- Bibliography

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Book SynopsisThis work contains a collection of lectures on stochastic processes. The material is arranged in such a way to fit a lecture time of one-and-a-half hours, in order to make the book convenient for lecturers and students. The book can be used in the preparation of courses in stochastic processes for which an understanding of basic notions of mathematical analysis, theory of complex functions, theory of differential equations and probability theory is required. The subjects in the book have different levels of abstraction.Table of ContentsFrontmatter -- Contents -- Preface -- Lecture 1. Stochastic processes. definitions. examples -- Lecture 2. The kolmogorov consistency theorem. classification of processes -- Lecture 3. Random walks. recurrence. renewal theorem -- Lecture 4. Martingales. inequalities for martingales -- Lecture 5. Theorems on the limit of a martingale -- Lecture 6. Stationary sequences. ergodic theorem -- Lecture 7. Ergodic theorem. metric transitivity -- Lecture 8. Regularization of a process. continuity -- Lecture 9. Processes without discontinuities of the second kind -- Lecture 10. Continuity of processes with independent increments. martingales with continuous time -- Lecture 11. Measurable processes -- Lecture 12. Stopping times. associated tr-algebras -- Lecture 13. Completely measurable processes -- Lecture 14. L2-theory -- Lecture 15. Stochastic integrals -- Lecture 16. Stationary processes. spectral representations -- Lecture 17. Stationary sequences. regularity and singularity -- Lecture 18. The prediction of a stationary sequence -- Lecture 19. Markov processes -- Lecture 20. Homogeneous markov processes and associated semigroups -- Lecture 21. Homogeneous purely discontinuous processes. conditions for their regularity -- Lecture 22. Processes with adenumerable set of states -- Lecture 23. Simple birth and death processes -- Lecture 24. Branching processes with particles of only one kind -- Lecture 25. Homogeneous processes and strongly continuous semigroups. resolvent operator and generator -- Lecture 26. The hille-iosida theorem -- Lecture 27. Processes with independent increments. representation of the discontinuous part -- Lecture 28. General representation of a stochastically continuous process with independent increments -- Lecture 29. Diffusion processes -- Lecture 30. Stochastic integrals -- Lecture 31. Existence, uniqueness, and properties of solutions of stochastic differential equations -- Lecture 32. Itô's formula with some corollaries -- Bibliography

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Book SynopsisThis book is designed for undergraduate programs and students and can also be used as a first-year graduate text in probability. It offers a broad perspective, building on the synopsis of measure and integration offered in Chapter two.

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Book SynopsisStructured in two parts, the first section focuses on probabilistic graphical models, while the second part deals with decision graphs, and in addition to the frameworks described in the previous edition, it also introduces Markov decision process and partially ordered decision problems.Trade ReviewFrom the reviews:MATHEMATICAL REVIEWS"This is indeed an invaluable text for students in information technology, engineering, and statistics. It is also very helpful for researchers in these fields and for those working in industry. The book is self-contained…The book has enough illustrative examples and exercises for the reader. All the illustrations are motivated by real applications. Moreover, the book provides a good balance between pure mathematical treatment and the applied aspects of the subject.""The Bayesian network (BN), or probabilistic expert system, is technology for automating human-life reasoning under uncertainty in specific contexts. … the book does an admirable job of concisely explaining a great range of concepts and techniques. … the book is very well written and to my knowledge nothing else meets its specific goal of quickly equipping the reader with both practical skills and sufficient theoretical background. … I certainly would not want to try to implement a BN application without reading this book.” (David Tritchler, Sankhya: Indian Journal of Statistics, Vol. 64 (B Part 3), 2002)"Professor Jensen is certainly one of the most influential researchers in the field of Bayesian networks and it is not surprising that this book represents a very clear and useful presentation of the main properties and use of graphical models. … I think that the present volume represents a useful integration of other material and a compact guide for either a student who wants an introduction to the field or a teacher who needs a reference for a course on probabilistic reasoning in AI." (Luigi Portinale, The Computer Journal, Vol. 46 (3), 2003)"This book is an introduction to Bayesian networks at an accessible level for first-year graduate or advanced undergraduate students. … I found this book to be an excellent introduction to the topic. It is well written, provides broad topic coverage, and is quite accessible to the non-expert. … I think Bayesian Networks and Decision Graphs would make a fine text for an introductory class in Bayesian networks or a useful reference for anyone interested in learning about the field." (David J. Marchette, Technometrics, Vol. 45 (2), 2003)"I can comfortably recommend this book as a primary source for topics related to Bayesian networks and decision graphs. This would be an excellent edition to any personal library." (Technometrics, Feburary 2008)From the reviews of the second edition:"The present book provides a very readable but also rigorous and comprehensive introduction to the subject. It would make a very good text for a graduate or an advanced undergraduate course. … Altogether, this is a very useful book for anyone interested in learning Bayesian networks without tears." (Jayanta K. Ghosh, International Statistical Reviews, Vol. 76 (2), 2008)"This book is the second edition of Jensen’s Bayesian Networks and Decision Graphs … . Each chapter ends with a summary section, bibliographic notes, and exercises. … provides a readable, self-contained, and above all, practical introduction to Bayesian networks and decision graphs. Its treatment is appropriate not just for statisticians, but also for computer scientists, engineers, and others researchers with appropriate mathematical background. … highly recommend it as a text or a useful reference for anyone interested in probabilistic graphical models or decision graphs." (Alyson G. Wilson, Journal of the American Statistical Association, Vol. 104 (485), March, 2009)“Devoted to Bayesian Networks or Graphical Models and Influence Diagrams, covering a full course with nice exercises … . It is useful as a reference for special topics. … strongly recommended for readers or user of BNs who are interested in specifying dependency models. … great importance to practitioners who try to find causality behind call-backs of products or crashes. … the book can be recommended to anybody working on the interface of operations research, AI, statistics and computer science.”­­­ (Hans-J. Lenz, Statistical Papers, Vol. 52, 2011)Table of ContentsCausal and Bayesian Networks * Part I: A Practical Guide to Normative Systems: Building Models * Learning, Adaptation, and Tuning * Decision Graphs * Part II: Algorithms for Normative Systems: Belief Updating in Bayesian Networks * Bayesian Network Analysis Tools * Algorithms for Influence Diagrams

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Book SynopsisI: The Dichotomous Rasch Model.- 1. Some Background for Item Response Theory and the Rasch Model.- 2. Derivations of the Rasch Model.- 3. Estimation of Item Parameters.- 4. On Person Parameter Estimation in the Dichotomous Rasch Model.- 5. Testing the Rasch Model.- 6. The Assessment of Person Fit.- 7. Test Construction from Item Banks.- II: Extensions of the Dichotomous Rasch Model.- 8. The Linear Logistic Test Model.- 9. Linear Logistic Models for Change.- 10. Dynamic Generalizations of the Rasch Model.- 11. Linear and Repeated Measures Models for the Person Parameters.- 12. The One Parameter Logistic Model.- 13. Linear Logistic Latent Class Analysis and the Rasch Model.- 14. Mixture Distribution Rasch Models.- III: Polytomous Rasch Models and their Extensions.- 15. Polytomous Rasch Models and their Estimation.- 16. The Derivation of Polytomous Rasch Models.- 17. The Polytomous Rasch Model within the Class of Generalized Linear Symmetry Models.- 18. Tests of Fit for Polytomous Rasch MTable of ContentsI: The Dichotomous Rasch Model.- 1. Some Background for Item Response Theory and the Rasch Model.- 2. Derivations of the Rasch Model.- 3. Estimation of Item Parameters.- 4. On Person Parameter Estimation in the Dichotomous Rasch Model.- 5. Testing the Rasch Model.- 6. The Assessment of Person Fit.- 7. Test Construction from Item Banks.- II: Extensions of the Dichotomous Rasch Model.- 8. The Linear Logistic Test Model.- 9. Linear Logistic Models for Change.- 10. Dynamic Generalizations of the Rasch Model.- 11. Linear and Repeated Measures Models for the Person Parameters.- 12. The One Parameter Logistic Model.- 13. Linear Logistic Latent Class Analysis and the Rasch Model.- 14. Mixture Distribution Rasch Models.- III: Polytomous Rasch Models and their Extensions.- 15. Polytomous Rasch Models and their Estimation.- 16. The Derivation of Polytomous Rasch Models.- 17. The Polytomous Rasch Model within the Class of Generalized Linear Symmetry Models.- 18. Tests of Fit for Polytomous Rasch Models.- 19. Extended Rating Scale and Partial Credit Models for Assessing Change.- 20. Polytomous Mixed Rasch Models.- In Retrospect.- 21. What Georg Rasch Would Have Thought about this Book.- References.- Author Index.- Abbreviations.

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Book SynopsisThe aim of this graduate textbook is to provide a comprehensive advanced course in the theory of statistics covering those topics in estimation, testing, and large sample theory which a graduate student might typically need to learn as preparation for work on a Ph.D.Trade ReviewFrom the reviews: "Another excellent book in theory of statistics is by Mark J. Schervish. … Readers will enjoy reading this book to see how differently the theory can be presented … . This well written book contains nine chapters and four appendices. ... Each chapter has both easy and challenging problems. The book is suitable for graduate level statistical theory courses. Examples and illustrations are well explained. I liked the author’s presentation, and learned a lot from the book. I highly recommend this book to theoretical statisticians." (Ramalingam Shanmugam, Journal of Statistical Computation and Simulation, Vol. 74 (11), November, 2004)Table of ContentsContent.- 1: Probability Models.- 1.1 Background.- 1.1.1 General Concepts.- 1.1.2 Classical Statistics.- 1.1.3 Bayesian Statistics.- 1.2 Exchangeability.- 1.2.1 Distributional Symmetry.- 1.2.2 Frequency arid Exchangeability.- 1.3 Parametric Models.- 1.3.1 Prior, Posterior, and Predictive Distributions.- 1.3.2 Improper Prior Distributions.- 1.3.3 Choosing Probability Distributions.- 1.4 DeFinetti’s Representation Theorem.- 1.4.1 Understanding the Theorems.- 1.4.2 The Mathematical Statements.- 1.4.3 Some Examples.- 1.5 Proofs of DeFinetti’s Theorem and Related Results*.- 1.5.1 Strong Law of Large Numbers.- 1.5.2 The Bernoulli Case.- 1.5.3 The General Finite Case*.- 1.5.4 The General Infinite Case.- 1.5.5 Formal Introduction to Parametric Models*.- 1.6 Infinite-Dimensional Parameters*.- 1.6.1 Dirichlet Processes.- 1.6.2 Tailfree Processes+.- 1.7 Problems.- 2: Sufficient Statistics.- 2.1 Definitions.- 2.1.1 Notational Overview.- 2.1.2 Sufficiency.- 2.1.3 Minimal and Complete Sufficiency.- 2.1.4 Ancillarity.- 2.2 Exponential Families of Distributions.- 2.2.1 Basic Properties.- 2.2.2 Smoothness Properties.- 2.2.3 A Characterization Theorem*.- 2.3 Information.- 2.3.1 Fisher Information.- 2.3.2 Kullback-Leibler Information.- 2.3.3 Conditional Information*.- 2.3.4 Jeffreys’ Prior*.- 2.4 Extremal Families*.- 2.4.1 The Main Results.- 2.4.2 Examples.- 2.4.3 Proofs+.- 2.5 Problems.- Chapte 3: Decision Theory.- 3.1 Decision Problems.- 3.1.1 Framework.- 3.1.2 Elements of Bayesian Decision Theory.- 3.1.3 Elements of Classical Decision Theory.- 3.1.4 Summary.- 3.2 Classical Decision Theory.- 3.2.1 The Role of Sufficient Statistics.- 3.2.2 Admissibility.- 3.2.3 James—Stein Estimators.- 3.2.4 Minimax Rules.- 3.2.5 Complete Classes.- 3.3 Axiomatic Derivation of Decision Theory*.- 3.3.1 Definitions and Axioms.- 3.2.2 Examples.- 3.3.3 The Main Theorems.- 3.3.4 Relation to Decision Theory.- 3.3.5 Proofs of the Main Theorems*.- 3.3.6 State-Dependent Utility*.- 3.4 Problems.- 4: Hypothesis Testing.- 4.1 Introduction.- 4.1.1 A Special Kind of Decision Problem.- 4.1.2 Pure Significance Tests.- 4.2 Bayesian Solutions.- 4.2.1 Testing in General.- 4.2.2 Bayes Factors.- 4.3 Most Powerful Tests.- 4.3.1 Simple Hypotheses and Alternatives.- 4.3.2 Simple Hypotheses, Composite Alternatives.- 4.3.3 One-Sided Tests.- 4.3.4 Two-Sided Hypotheses.- 4.4 Unbiased Tests.- 4.4.1 General Results.- 4.4.2 Interval Hypotheses.- 4.4.3 Point Hypotheses.- 4.5 Nuisance Parameters.- 4.5.1 Neyinan Structure.- 4.5.2 Tests about Natural Parameters.- 4.5.3 Linear Combinations of Natural Parameters.- 4.5.4 Other Two-Sided Cases*.- 4.5.5 Likelihood Ratio Tests.- 4.5.6 The Standard F-Test as a Bayes Rule.- 4.6 P-Values.- 4.6.1 Definitions and Examples.- 4.6.2 P-Values and Bayes Factors.- 4.7 Problems.- 5: Estimation.- 5.1 Point Estimation.- 5.1.1 Minimum Variance Unbiased Estimation.- 5.1.2 Lower Bounds on the Variance of Unbiased Estimators.- 5.1.3 Maximum Likelihood Estimation.- 5.1.4 Bayesian Estimation.- 5.1.5 Robust Estimation*.- 5.2 Set Estimation.- 5.2.1 Confidence Sets.- 5.2.2 Prediction Sets*.- 5.2.3 Tolerance Sets*.- 5.2.4 Bayesian Set Estimation.- 5.2.5 Decision Theoretic Set Estimation.- 5.3 The Bootstrap*.- 5.3.1 The General Concept.- 5.3.2 Standard Deviations and Bias.- 5.3.3 Bootstrap Confidence Intervals.- 5.4 Problems.- 6: Equivariance*.- 6.1 Common Examples.- 6.1.1 Location Problems.- 6.1.2 Scale Problems.- 6.2 Equivariant Decision Theory.- 6.2.1 Groups of Transformations.- 6.2.2 Equivariance and Changes of Units.- 6.2.3 Minimum Risk Equivariant Decisions.- 6.3 Testing and Confidence Intervals*.- 6.3.1 P-Values in Invariant Problems.- 6.3.2 Equivariant Confidence Sets.- 6.3.3 Invariant Tests*.- 6.4 Problems.- 7: Large Sample Theory.- 7.1 Convergence Concepts.- 7.1.1 Deterministic Convergence.- 7.1.2 Stochastic Convergence.- 7.1.3 The Delta Method.- 7.2 Sample Quantiles.- 7.2.1 A Single Quantile.- 7.2.2 Several Quantiles.- 7.2.3 Linear Combinations of Quantiles*.- 7.3 Large Sample Estimation.- 7.3.1 Some Principles of Large Sample Estimation.- 7.3.2 Maximum Likelihood Estimators.- 7.3.3 MLEs in Exponential Families.- 7.3.4 Examples of Inconsistent MLEs.- 7.3.5 Asymptotic Normality of MLEs.- 7.3.6 Asymptotic Properties of M-Estimators.- 7.4 Large Sample Properties of Posterior Distributions.- 7.4.1 Consistency of Posterior Distributions+.- 7.4.2 Asymptotic Normality of Posterior Distributions.- 7.4.3 Laplace Approximations to Posterior Distributions*.- 7.4.4 Asymptotic Agreement of Predictive Distributions+.- 7.5 Large Sample Tests.- 7.5.1 Likelihood Ratio Tests.- 7.5.2 Chi-Squarcd Goodness of Fit Tests.- 7.6 Problems.- 8: Hierarchical Models.- 8.1 Introduction.- 8.1.1 General Hierarchical Models.- 8.1.2 Partial Exchangeability*.- 8.1.3 Examples of the Representation Theorem*.- 8.2 Normal Linear Models.- 8.2.1 One-Way ANOVA.- 8.2.2 Two-Way Mixed Model ANOVA*.- 8.2.3 Hypothesis Testing.- 8.3 Nonnormal Models*.- 8.3.1 Poisson Process Data.- 8.3.2 Bernoulli Process Data.- 8.4 Empirical Bayes Analysis*.- 8.4.1 Naïve Empirical Bayes.- 8.4.2 Adjusted Empirical Bayes.- 8.4.3 Unequal Variance Case.- 8.5 Successive Substitution Sampling.- 8.5.1 The General Algorithm.- 8.5.2 Normal Hierarchical Models.- 8.5.3 Nonnormal Models.- 8.6 Mixtures of Models.- 8.6.1 General Mixture Models.- 8.6.2 Outliers.- 8.6.3 Bayesian Robustness.- 8.7 Problems.- 9: Sequential Analysis.- 9.1 Sequential Decision Problems.- 9.2 The Sequential Probability Ratio Test.- 9.3 Interval Estimation*.- 9.4 The Relevancc of Stopping Rules.- 9.5 Problems.- Appendix A: Measure and Integration Theory.- A.1 Overview.- A.1.1 Definitions.- A.1.2 Measurable Functions.- A.1.3 Integration.- A.1.4 Absolute Continuity.- A.2 Measures.- A.3 Measurable Functions.- A.4 Integration.- A.5 Product Spaces.- A.6 Absolute Continuity.- A.7 Problems.- Appendix B: Probability Theory.- B.1 Overview.- B.1.1 Mathematical Probability.- B.1.2 Conditioning.- B.1.3 Limit Theorems.- B.2 Mathematical Probability.- B.2.1 Random Quantities and Distributions.- B.2.2 Some Useful Inequalities.- B.3 Conditioning.- B.3.1 Conditional Expectations.- B.3.2 Borel Spaces*.- B.3.3 Conditional Densities.- B.3.4 Conditional Independence.- B.3.5 The Law of Total Probability.- B.4 Limit Theorems.- B.4.1 Convergence in Distribution and in Probability.- B.4.2 Characteristic Functions.- B.5 Stochastic Processes.- B.5.1 Introduction.- B.5.3 Markov Chains*.- B.5.4 General Stochastic Processes.- B.6 Subjective Probability.- B.7 Simulation*.- B.8 Problems.- Appendix C: Mathematical Theorems Not Proven Here.- C.1 Real Analysis.- C.2 Complex Analysis.- C.3 Functional Analysis.- Appendix D: Summary of Distributions.- D.1 Univariate Continuous Distributions.- D.2 Univariate Discrete Distributions.- D.3 Multivariate Distributions.- References.- Notation and Abbreviation Index.- Name Index.

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Book SynopsisExamples.- A Model for Longitudinal Data.- Exploratory Data Analysis.- Estimation of the Marginal Model.- Inference for the Marginal Model.- Inference for the Random Effects.- Fitting Linear Mixed Models with SAS.- General Guidelines for Model Building.- Exploring Serial Correlation.- Local Influence for the Linear Mixed Model.- The Heterogeneity Model.- Conditional Linear Mixed Models.- Exploring Incomplete Data.- Joint Modeling of Measurements and Missingness.- Simple Missing Data Methods.- Selection Models.- Pattern-Mixture Models.- Sensitivity Analysis for Selection Models.- Sensitivity Analysis for Pattern-Mixture Models.- How Ignorable Is Missing At Random ?.- The Expectation-Maximization Algorithm.- Design Considerations.- Case Studies.Trade ReviewFrom the reviews: MATHEMATICAL REVIEWS "This book emphasizes practice rather than mathematical rigor and the majority of the chapters are explanatory rather than research oriented. In this respect, guidance and advice on practical issues are the main focus of the text. Hence it will be of interest to applied statisticians and biomedical researchers in industry, particularly in the pharmaceutical industry, medical public health organizations, contract research organizations, and academia." "This book provides a comprehensive treatment of linear mixed models for continuous longitudinal data. Over 125 illustrations are included in the book. … I do believe that the book may serve as a useful reference to a broader audience. Since practical examples are provided as well as discussion of the leading software utilization, it may also be appropriate as a textbook in an advanced undergraduate-level or a graduate-level course in an applied statistics program." (Ana Ivelisse Avil és, Technometrics, Vol. 43 (3), 2001) "A practical book with a great many examples, including worked computer code and access to the datasets. … The authors state that the book covers ‘linear mixed models for continuous outcomes’ … . The book has four main strengths: its practical bent, its emphasis on exploratory analysis, its description of tools for model checking, and its treatment of dropout and missingness … . my impression of the book was … positive. Its strong practical nature and emphasis on dropout modelling are particularly welcome … ." (Harry Southworth, ISCB Newsletter, June, 2002) "This book is devoted to linear mixed-effects models with strong emphasis on the SAS procedure. Guidance and advice on practical issues are the main focus of the text. … It is of value to applied statisticians and biomedical researchers. … I recommend this book as a reference to applied statisticians and biomedical researchers, particularly in the pharmaceutical industry, medical and public organizations." (Wang Songgui, Zentralblatt MATH, Vol. 956, 2001)Table of ContentsIntroduction * Examples * A model for Longitudinal Data * Exploratory Data Analysis * Estimation of the Marginal Model * Inference for the Marginal Model * Inference for the Random Effects * Fitting Linear Mixed Models with SAS * General Guidelines for Model Building * Exploring Serial Correlation * Local Influence for the Linear Mixed Model * The Heterogeneity Model * Conditional Linear Mixed Models * Exploring Incomplete Data * Joint Modeling of Measurements and Missingness * Simple Missing Data Methods * Selection Models * Pattern-Mixture Models * Sensitivity Analysis for Selection Models * Sensitivity Analysis for Models * How Ignorable is Missing at Random? * The Expectation-Maximization Algorithm * Design Considerations * Case Studies

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Book SynopsisA comprehensive introduction to a wide variety of statistical methods for the analysis of repeated measurements. It is designed to be both a useful reference for practitioners and a textbook for a graduate-level course focused on methods for the analysis of repeated measurements.Trade ReviewFrom the reviews: MATHEMATICAL REVIEWS "…the book covers a wide range of topics, including inference based on normal theory, repeated categorical outcomes and missing values. The book is based on lecture notes used by the author since 1991. Hence, the material and the structure of the book have been well tested by different audiences. Another feature of the book is the inclusion of a very rich collection of problems with excellent real data. Thus, it is a nice textbook for a semester course on repeated measurements and longitudinal data." SHORT BOOK REVIEW "Each major topic is introduced logically; background theory is clearly elucidated, and at least one example is carefully worked in detail. The use of eighty real sets of data, given in full, is a most attractive feature. Attention is concentrated on those techniques that are most readily available in software. ... This should prove to be a very useful text for teacher, student and practitioner alike." JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION "Most other books on repeated measurements tend to focus on specialized topics. In my opinion, [this] book is the most comprehensive and readable of the lot. I would highly recommend its use as a text for a semester-length graduate course for biostatistics and statistics students and also as resource book for consulting biostatisticians and statisticians. In addition, this book would be a valuable resource for students from other fields of study (e.g., the health sciences) who have a statistical aptitude. The book is definitely worth the price." "The intention of the book is ‘to provide a reasonably comprehensive overview of methods for the analysis of repeated measurements’ with focus on standard statistical methods … . In my opinion the book gives a nice, comprehensive overview of methods for the analysis of repeated measurements. … The availability of data sets, overheads, etc. is a very valuable supplement for both teachers and students. … The book … could be a natural choice for a course in repeated measurements for graduate students in (bio-) statistics." (Niels Trolle Andersen, Statistics in Medicine, Vol. 24 (5), 2005) "This book is a very interesting and comprehensive summary of a wide selection of statistical methods for the analysis of repeated measurements. It is indeed an ideal and carefully written text to be used as a reference guide for practitioners and, in addition, as a great, up to date and very complete textbook for a graduate-level course in Statistics and/or Biostatistics. … I highly recommend Statistical Methods … as a good reference book for anyone interested in looking into the different available methodologies … ." (Vicente Núñez-Antón, Journal of Applied Statistics, Vol. 30 (10), December, 2003) "This book provides a comprehensive introduction to a wide variety of statistical methods for the analysis of repeated measurements. … In conclusion, as a course text on repeated measurements this book clearly has major strengths over others in that it provides coverage on a wide range of topics and provides extensive further reading material. … I would recommend this text as a general reference book on repeated measurements which would make a worthwhile addition to a departmental library." (Fiona Holland, Pharmaceutical Statistics, 2003) "Most other books on repeated measurements … tend to focus on specialized topics. In my opinion, Statistical Methods for the Analysis of Repeated Measurements book is the most comprehensive and readable of the lot. I would highly recommend its use as a text for a semester-length graduate course for biostatistics and statistics students and … for consulting biostatisticians and statisticians. … a valuable resource for students from other fields of study … who have a statistical aptitude. The book is definitely worth the price." (Melvin L. Moeschberger, Journal of the American Statistical Association, March, 2003) "The book aims at describing, discussing and demonstrating a variety of statistical methods for the analysis of repeated measurements … . the book covers a very wide range of topics, including inference based on normal theory, repeated categorical outcomes and missing values. … Another feature of the book is the inclusion of a very rich collection of problems with excellent real data. Thus, it is a nice textbook for a semester course on repeated measurements and longitudinal data." (Jack C. Lee, Mathematical Reviews, 2003 e) "This book is intended to provide a comprehensive introduction to a wide range of statistical methods for the analysis of repeated measurements. … For use in a course, I would use it for an applied graduate-level statistics course on linear models for analysis of repeated measurements. This text is useful not only with regards to the statistical methods, but also for the real data examples that can be explored with the various models and methods under study." (James R. Kenyon, Technometrics, Vol. 45 (1), 2003) "This book provides a reasonably comprehensive overview of a wide variety of statistical methods for the analysis of repeated measurements … . The important features of this book include a summary of both classical and recent methods for continuous and categorical outcome variables, numerous homework problems, and the extensive use of real data sets in examples. … This book will be of interest to graduate students in statistics and biostatistics as well as to practicing statisticians in academic, industry and research institutions.” (Ivan Krivý, Zentralblatt MATH, Vol. 985, 2002)Table of ContentsIntroduction * Univariate Methods * Normal-Theory Methods: Unstructured Multivariate Approach * Normal Theory Methods: Multivariate Analysis of Variance * Normal-Theory Methods: Repeated Measures ANOVA * Normal Theory Methods: Linear Mixed Models * Weighted Least Squares Analysis of Repeated Categorical Outcomes * Randomization Model Methods for One-Sample Repeated Measurements * Methods Based on Extensions of Generalized Linear Models * Nonparametric Methods

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Book SynopsisPoint processes and random measures find wide applicability in telecommunications, earthquakes, image analysis, spatial point patterns, and stereology. This volume relates to marked point processes and to processes evolving in time, where the conditional intensity methodology provides a basis for model building, inference, and prediction.Trade ReviewFrom the reviews of the second edition: "This is an important treatise on the mathematical theory relevant to a wide variety of random processes…The reader will find excellent treatments of important advanced topics such as Cox, renewal, Wold, marked, cluster, and other specialized processes, plus concise but useful appendices on topology, measure theory, metric spaces, martingales, and the like." Technometrics, May 2004 "This revision splits the lengthy first version into two volumes now subtitled Volume I: Elementary Theory and Methods and Volume II: Models and General Theory and Structure...this first volume is well worth its price." Journal of the American Statistical Association, September 2004 "The theory of point processes has undergone an explosive expansion in the last two decades. There was a genuine need for a single source that would contain a survey of the general theory of Point Process accessible to beginning graduate students and researchers in the field but at the same time would include some of the more recent – though relatively advanced and technically difficult – developments in the area. The present edition of the book addresses that need quite successfully.” (Alok Goswami, Sankhya, Vol. 67 (1), 2005) "It is a pleasure to announce that a second edition of the classic book ‘An Introduction to the Theory of Point Processes’ has been published. … many chapters and sections were thoroughly reworked. This holds true in particular for the exercises, which are obviously produced with particular love. … The reviewer is sure that the owners of the first edition will buy also the second, and for many younger readers it will become the beloved key reference to point processes." (Dietrich Stoyan, Metrika, May, 2004) "The second edition of this monograph is divided into two volumes. The first one is concentrated on introductory material and models, the second one on structure and general theory. … suitable as a textbook with many exercises for beginners as well as a source for scientists interested in high level applications of point processes." (Uwe Küchler, Zentralblatt MATH, Vol. 1026, 2004) "The first edition of this book by two major research workers in the field speedily established itself as an authoritative account of an important and rapidly developing subject. In this substantially revised and expanded second edition, the authors have wisely decided to divide the book into two parts leaving some of the very technical material … to a second volume. … The book is likely to establish itself quickly as a major contribution to the field." (D. R. Cox, Short Book Reviews, Vol. 23 (2), 2003)Table of ContentsEarly History * Basic Properties of the Poisson Process * Simple Results for Stationary Point Processes on the Line * Renewal Processes * Finite Point Processes * Models Constructed via Conditioning: Cox, Cluster,and Marked Point Processes * Conditional Intensities and Likelihoods * Second Order Properties of Stationary Point Processes

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Book Synopsis1 Controlled Markov Processes.- 1.1 Introduction.- 1.2 Stochastic Control Problems.- 1.3 Examples.- 1.4 Further Comments.- 2 Discounted Reward Criterion.- 2.1 Introduction.- 2.2 Optimality Conditions.- 2.3 Asymptotic Discount Optimality.- 2.4 Approximation of MCM's.- 2.5 Adaptive Control Models.- 2.6 Nonparametric Adaptive Control.- 2.7 Comments and References.- 3 Average Reward Criterion.- 3.1 Introduction.- 3.2 The Optimality Equation.- 3.3 Ergodicity Conditions.- 3.4 Value Iteration.- 3.5 Approximating Models.- 3.6 Nonstationary Value Iteration.- 3.7 Adaptive Control Models.- 3.8 Comments and References.- 4 Partially Observable Control Models.- 4.1 Introduction.- 4.2 PO-CM: Case of Known Parameters.- 4.3 Transformation into a CO Control Problem.- 4.4 Optimal I-Policies.- 4.5 PO-CM's with Unknown Parameters.- 4.6 Comments and References.- 5 Parameter Estimation in MCM's.- 5.1 Introduction.- 5.2 Contrast Functions.- 5.3 Minimum Contrast Estimators.- 5.4 Comments and References.- 6 Discretization Procedures.- 6.1 Introduction.- 6.2 Preliminaries.- 6.3 The Non-Adaptive Case.- 6.4 Adaptive Control Problems.- 6.5 Proofs.- 6.6 Comments and References.- Appendix A. Contraction Operators.- Appendix B. Probability Measures.- Total Variation Norm.- Weak Convergence.- Appendix C. Stochastic Kernels.- Appendix D. Multifunctions and Measurable Selectors.- The Hausdorff Metric.- Multifunctions.- References.- Author Index.Table of Contents1 Controlled Markov Processes.- 1.1 Introduction.- 1.2 Stochastic Control Problems.- Control Models.- Policies.- Performance Criteria.- Control Problems.- 1.3 Examples.- An Inventory/Production System.- Control of Water Reservoirs.- Fisheries Management.- Nonstationary MCM’s.- Semi-Markov Control Models.- 1.4 Further Comments.- 2 Discounted Reward Criterion.- 2.1 Introduction.- Summary.- 2.2 Optimality Conditions.- Continuity of ?*.- 2.3 Asymptotic Discount Optimality.- 2.4 Approximation of MCM’s.- Nonstationary Value-Iteration.- Finite-State Approximations.- 2.5 Adaptive Control Models.- Preliminaries.- Nonstationary Value-Iteration.- The Principle of Estimation and Control.- Adaptive Policies.- 2.6 Nonparametric Adaptive Control.- The Parametric Approach.- New Setting.- The Empirical Distribution Process.- Nonparametric Adaptive Policies.- 2.7 Comments and References.- 3 Average Reward Criterion.- 3.1 Introduction.- Summary.- 3.2 The Optimality Equation.- 3.3 Ergodicity Conditions.- 3.4 Value Iteration.- Uniform Approximations.- Successive Averagings.- 3.5 Approximating Models.- 3.6 Nonstationary Value Iteration.- Nonstationary Successive Averagings.- Discounted-Like NVI.- 3.7 Adaptive Control Models.- Preliminaries.- The Principle of Estimation and Control (PEC).- Nonstationary Value Iteration (NVI).- 3.8 Comments and References.- 4 Partially Observable Control Models.- 4.1 Introduction.- Summary.- 4.2 PO-CM: Case of Known Parameters.- The PO Control Problem.- 4.3 Transformation into a CO Control Problem.- I-Policies.- The New Control Model.- 4.4 Optimal I-Policies.- 4.5 PO-CM’s with Unknown Parameters.- PEC and NVI I-Policies.- 4.6 Comments and References.- 5 Parameter Estimation in MCM’s.- 5.1 Introduction.- Summary.- 5.2 Contrast Functions.- 5.3 Minimum Contrast Estimators.- 5.4 Comments and References.- 6 Discretization Procedures.- 6.1 Introduction.- Summary.- 6.2 Preliminaries.- 6.3 The Non-Adaptive Case.- A Non-Recursive Procedure.- A Recursive Procedure.- 6.4 Adaptive Control Problems.- Preliminaries.- Discretization of the PEC Adaptive Policy.- Discretization of the NVI Adaptive Policy.- 6.5 Proofs.- The Non-Adaptive Case.- The Adaptive Case.- 6.6 Comments and References.- Appendix A. Contraction Operators.- Appendix B. Probability Measures.- Total Variation Norm.- Weak Convergence.- Appendix C. Stochastic Kernels.- Appendix D. Multifunctions and Measurable Selectors.- The Hausdorff Metric.- Multifunctions.- References.- Author Index.

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Book Synopsis1 Martingales, Stopping Times, and Filtrations.- 1.1. Stochastic Processes and ?-Fields.- 1.2. Stopping Times.- 1.3. Continuous-Time Martingales.- 1.4. The DoobMeyer Decomposition.- 1.5. Continuous, Square-Integrable Martingales.- 1.6. Solutions to Selected Problems.- 1.7. Notes.- 2 Brownian Motion.- 2.1. Introduction.- 2.2. First Construction of Brownian Motion.- 2.3. Second Construction of Brownian Motion.- 2.4. The SpaceC[0, ?), Weak Convergence, and Wiener Measure.- 2.5. The Markov Property.- 2.6. The Strong Markov Property and the Reflection Principle.- 2.7. Brownian Filtrations.- 2.8. Computations Based on Passage Times.- 2.9. The Brownian Sample Paths.- 2.10. Solutions to Selected Problems.- 2.11. Notes.- 3 Stochastic Integration.- 3.1. Introduction.- 3.2. Construction of the Stochastic Integral.- 3.3. The Change-of-Variable Formula.- 3.4. Representations of Continuous Martingales in Terms of Brownian Motion.- 3.5. The Girsanov Theorem.- 3.6. Local Time and a Generalized Itô Rule for Brownian Motion.- 3.7. Local Time for Continuous Semimartingales.- 3.8. Solutions to Selected Problems.- 3.9. Notes.- 4 Brownian Motion and Partial Differential Equations.- 4.1. Introduction.- 4.2. Harmonic Functions and the Dirichlet Problem.- 4.3. The One-Dimensional Heat Equation.- 4.4. The Formulas of Feynman and Kac.- 4.5. Solutions to selected problems.- 4.6. Notes.- 5 Stochastic Differential Equations.- 5.1. Introduction.- 5.2. Strong Solutions.- 5.3. Weak Solutions.- 5.4. The Martingale Problem of Stroock and Varadhan.- 5.5. A Study of the One-Dimensional Case.- 5.6. Linear Equations.- 5.7. Connections with Partial Differential Equations.- 5.8. Applications to Economics.- 5.9. Solutions to Selected Problems.- 5.10. Notes.- 6 P. Lévy's Theory of Brownian Local Time.-6.1. Introduction.- 6.2. Alternate Representations of Brownian Local Time.- 6.3. Two Independent Reflected Brownian Motions.- 6.4. Elastic Brownian Motion.- 6.5. An Application: Transition Probabilities of Brownian Motion with Two-Valued Drift.- 6.6. Solutions to Selected Problems.- 6.7. Notes.Trade ReviewSecond Edition I. Karatzas and S.E. Shreve Brownian Motion and Stochastic Calculus "A valuable book for every graduate student studying stochastic process, and for those who are interested in pure and applied probability. The authors have done a good job."—MATHEMATICAL REVIEWSTable of Contents1 Martingales, Stopping Times, and Filtrations.- 1.1. Stochastic Processes and ?-Fields.- 1.2. Stopping Times.- 1.3. Continuous-Time Martingales.- A. Fundamental inequalities.- B. Convergence results.- C. The optional sampling theorem.- 1.4. The Doob—Meyer Decomposition.- 1.5. Continuous, Square-Integrable Martingales.- 1.6. Solutions to Selected Problems.- 1.7. Notes.- 2 Brownian Motion.- 2.1. Introduction.- 2.2. First Construction of Brownian Motion.- A. The consistency theorem.- B. The Kolmogorov—?entsov theorem.- 2.3. Second Construction of Brownian Motion.- 2.4. The SpaceC[0, ?), Weak Convergence, and Wiener Measure.- A. Weak convergence.- B. Tightness.- C. Convergence of finite-dimensional distributions.- D. The invariance principle and the Wiener measure.- 2.5. The Markov Property.- A. Brownian motion in several dimensions.- B. Markov processes and Markov families.- C. Equivalent formulations of the Markov property.- 2.6. The Strong Markov Property and the Reflection Principle.- A. The reflection principle.- B. Strong Markov processes and families.- C. The strong Markov property for Brownian motion.- 2.7. Brownian Filtrations.- A. Right-continuity of the augmented filtration for a strong Markov process.- B. A “universal” filtration.- C. The Blumenthal zero-one law.- 2.8. Computations Based on Passage Times.- A. Brownian motion and its running maximum.- B. Brownian motion on a half-line.- C. Brownian motion on a finite interval.- D. Distributions involving last exit times.- 2.9. The Brownian Sample Paths.- A. Elementary properties.- B. The zero set and the quadratic variation.- C. Local maxima and points of increase.- D. Nowhere differentiability.- E. Law of the iterated logarithm.- F. Modulus of continuity.- 2.10. Solutions to Selected Problems.- 2.11. Notes.- 3 Stochastic Integration.- 3.1. Introduction.- 3.2. Construction of the Stochastic Integral.- A. Simple processes and approximations.- B. Construction and elementary properties of the integral.- C. A characterization of the integral.- D. Integration with respect to continuous, local martingales.- 3.3. The Change-of-Variable Formula.- A. The Itô rule.- B. Martingale characterization of Brownian motion.- C. Bessel processes, questions of recurrence.- D. Martingale moment inequalities.- E. Supplementary exercises.- 3.4. Representations of Continuous Martingales in Terms of Brownian Motion.- A. Continuous local martingales as stochastic integrals with respect to Brownian motion.- B. Continuous local martingales as time-changed Brownian motions.- C. A theorem of F. B. Knight.- D. Brownian martingales as stochastic integrals.- E. Brownian functionals as stochastic integrals.- 3.5. The Girsanov Theorem.- A. The basic result.- B. Proof and ramifications.- C. Brownian motion with drift.- D. The Novikov condition.- 3.6. Local Time and a Generalized Itô Rule for Brownian Motion.- A. Definition of local time and the Tanaka formula.- B. The Trotter existence theorem.- C. Reflected Brownian motion and the Skorohod equation.- D. A generalized Itô rule for convex functions.- E. The Engelbert—Schmidt zero-one law.- 3.7. Local Time for Continuous Semimartingales.- 3.8. Solutions to Selected Problems.- 3.9. Notes.- 4 Brownian Motion and Partial Differential Equations.- 4.1. Introduction.- 4.2. Harmonic Functions and the Dirichlet Problem.- A. The mean-value property.- B. The Dirichlet problem.- C. Conditions for regularity.- D. Integral formulas of Poisson.- E. Supplementary exercises.- 4.3. The One-Dimensional Heat Equation.- A. The Tychonoff uniqueness theorem.- B. Nonnegative solutions of the heat equation.- C. Boundary crossing probabilities for Brownian motion.- D. Mixed initial/boundary value problems.- 4.4. The Formulas of Feynman and Kac.- A. The multidimensional formula.- B. The one-dimensional formula.- 4.5. Solutions to selected problems.- 4.6. Notes.- 5 Stochastic Differential Equations.- 5.1. Introduction.- 5.2. Strong Solutions.- A. Definitions.- B. The Itô theory.- C. Comparison results and other refinements.- D. Approximations of stochastic differential equations.- E. Supplementary exercises.- 5.3. Weak Solutions.- A. Two notions of uniqueness.- B. Weak solutions by means of the Girsanov theorem.- C. A digression on regular conditional probabilities.- D. Results of Yamada and Watanabe on weak and strong solutions.- 5.4. The Martingale Problem of Stroock and Varadhan.- A. Some fundamental martingales.- B. Weak solutions and martingale problems.- C. Well-posedness and the strong Markov property.- D. Questions of existence.- E. Questions of uniqueness.- F. Supplementary exercises.- 5.5. A Study of the One-Dimensional Case.- A. The method of time change.- B. The method of removal of drift.- C. Feller’s test for explosions.- D. Supplementary exercises.- 5.6. Linear Equations.- A. Gauss—Markov processes.- B. Brownian bridge.- C. The general, one-dimensional, linear equation.- D. Supplementary exercises.- 5.7. Connections with Partial Differential Equations.- A. The Dirichlet problem.- B. The Cauchy problem and a Feynman—Kac representation.- C. Supplementary exercises.- 5.8. Applications to Economics.- A. Portfolio and consumption processes.- B. Option pricing.- C. Optimal consumption and investment (general theory).- D. Optimal consumption and investment (constant coefficients).- 5.9. Solutions to Selected Problems.- 5.10. Notes.- 6 P. Lévy’s Theory of Brownian Local Time.- 6.1. Introduction.- 6.2. Alternate Representations of Brownian Local Time.- A. The process of passage times.- B. Poisson random measures.- C. Subordinators.- D. The process of passage times revisited.- E. The excursion and downcrossing representations of local time.- 6.3. Two Independent Reflected Brownian Motions.- A. The positive and negative parts of a Brownian motion.- B. The first formula of D. Williams.- C. The joint density of (W(t), L(t), ? +(t)).- 6.4. Elastic Brownian Motion.- A. The Feynman—Kac formulas for elastic Brownian motion.- B. The Ray—Knight description of local time.- C. The second formula of D. Williams.- 6.5. An Application: Transition Probabilities of Brownian Motion with Two-Valued Drift.- 6.6. Solutions to Selected Problems.- 6.7. Notes.

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Book SynopsisComputational inference is based on an approach to statistical methods that uses modern computational power to simulate distributional properties of estimators and test statistics.Trade ReviewFrom the reviews:“This is a book that covers many of the computational issues that statisticians will encounter as part of their research and applied work. … The writing in the book is quite clear and the author has done a good job providing the essence of each topic. … Overall, I think this is an excellent book. … This book will give a graduate student a good overview of the field. There are exercises provided for each chapter together with some solutions.” (Michael J. Evans, Mathematical Reviews, Issue 2011 b)“This book is a superior treatment of the important subject of statistical computing. I strongly recommend this book to anyone who analyzes data using either a commercial statistical software package or statistical computer programs written by the user or someone else. Thus this book is important not only for data oriented statisticians but for econometricians, psychometricians, political methodologists and biometricians as well. … All terms in this work including computing terms are clearly defined.” (Melvin Hinich, Technometrics, Vol. 53 (1), February, 2011)“I greatly appreciated the author’s command of both numerical and statistical computing … . The book also contains many exercises that substantiate the concepts, with solutions and hints in the appendix, an extensive bibliography, and a link to further literature and notes. The target readership includes undergraduates, postgraduates in statistics and allied fields such as computer science and mathematics, scientific research workers, and practitioners of statistics and numerical techniques. … I strongly recommend it for all scientific libraries.” (Soubhik Chakraborty, ACM Computing Reviews, October, 2010)“This book has a very large scope in that … it covers the dual fields of computational statistics and of statistical computing. … must-read for all students and researchers engaging into any kind of serious statistical programming. … is well-written, in a lively and personal style. … a reference book that should appear in the shortlist of any computational statistics/statistical computing graduate course as well as on the shelves of any researchers supporting his or her statistical practice with a significant dose of computing backup.”­­­ (Christian P. Robert, Statistical and Computation, Vol. 21, 2011)Table of ContentsPreliminaries.- Mathematical and Statistical Preliminaries.- Statistical Computing.- Computer Storage and Arithmetic.- Algorithms and Programming.- Approximation of Functions and Numerical Quadrature.- Numerical Linear Algebra.- Solution of Nonlinear Equations and Optimization.- Generation of Random Numbers.- Methods of Computational Statistics.- Graphical Methods in Computational Statistics.- Tools for Identification of Structure in Data.- Estimation of Functions.- Monte Carlo Methods for Statistical Inference.- Data Randomization, Partitioning, and Augmentation.- Bootstrap Methods.- Exploring Data Density and Relationships.- Estimation of Probability Density Functions Using Parametric Models.- Nonparametric Estimation of Probability Density Functions.- Statistical Learning and Data Mining.- Statistical Models of Dependencies.

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Book SynopsisIn the statement of a Pontryagin-type maximum principle there is an adjoint equation, which is an ordinary differential equation (ODE) in the (finite-dimensional) deterministic case and a stochastic differential equation (SDE) in the stochastic case.Trade ReviewFrom the reviews: SIAM REVIEW "The presentation of this book is systematic and self-contained…Summing up, this book is a very good addition to the control literature, with original features not found in other reference books. Certain parts could be used as basic material for a graduate (or postgraduate) course…This book is highly recommended to anyone who wishes to study the relationship between Pontryagin’s maximum principle and Bellman’s dynamic programming principle applied to diffusion processes." MATHEMATICS REVIEW This is an authoratative book which should be of interest to researchers in stochastic control, mathematical finance, probability theory, and applied mathematics. Material out of this book could also be used in graduate courses on stochastic control and dynamic optimization in mathematics, engineering, and finance curricula. Tamer Basar, Math. ReviewTable of Contents1. Basic Stochastic Calculus.- 1. Probability.- 1.1. Probability spaces.- 1.2. Random variables.- 1.3. Conditional expectation.- 1.4. Convergence of probabilities.- 2. Stochastic Processes.- 2.1. General considerations.- 2.2. Brownian motions.- 3. Stopping Times.- 4. Martingales.- 5. Itô’s Integral.- 5.1. Nondifferentiability of Brownian motion.- 5.2. Definition of Itô’s integral and basic properties.- 5.3. Itô’s formula.- 5.4. Martingale representation theorems.- 6. Stochastic Differential Equations.- 6.1. Strong solutions.- 6.2. Weak solutions.- 6.3. Linear SDEs.- 6.4. Other types of SDEs.- 2. Stochastic Optimal Control Problems.- 1. Introduction.- 2. Deterministic Cases Revisited.- 3. Examples of Stochastic Control Problems.- 3.1. Production planning.- 3.2. Investment vs. consumption.- 3.3. Reinsurance and dividend management.- 3.4. Technology diffusion.- 3.5. Queueing systems in heavy traffic.- 4. Formulations of Stochastic Optimal Control Problems.- 4.1. Strong formulation.- 4.2. Weak formulation.- 5. Existence of Optimal Controls.- 5.1. A deterministic result.- 5.2. Existence under strong formulation.- 5.3. Existence under weak formulation.- 6. Reachable Sets of Stochastic Control Systems.- 6.1. Nonconvexity of the reachable sets.- 6.2. Noncloseness of the reachable sets.- 7. Other Stochastic Control Models.- 7.1. Random duration.- 7.2. Optimal stopping.- 7.3. Singular and impulse controls.- 7.4. Risk-sensitive controls.- 7.5. Ergodic controls.- 7.6. Partially observable systems.- 8. Historical Remarks.- 3. Maximum Principle and Stochastic Hamiltonian Systems.- 1. Introduction.- 2. The Deterministic Case Revisited.- 3. Statement of the Stochastic Maximum Principle.- 3.1. Adjoint equations.- 3.2. The maximum principle and stochastic Hamiltonian systems.- 3.3. A worked-out example.- 4. A Proof of the Maximum Principle.- 4.1. A moment estimate.- 4.2. Taylor expansions.- 4.3. Duality analysis and completion of the proof.- 5. Sufficient Conditions of Optimality.- 6. Problems with State Constraints.- 6.1. Formulation of the problem and the maximum principle.- 6.2. Some preliminary lemmas.- 6.3. A proof of Theorem 6.1.- 7. Historical Remarks.- 4. Dynamic Programming and HJB Equations.- 1. Introduction.- 2. The Deterministic Case Revisited.- 3. The Stochastic Principle of Optimality and the HJB Equation.- 3.1. A stochastic framework for dynamic programming.- 3.2. Principle of optimality.- 3.3. The HJB equation.- 4. Other Properties of the Value Function.- 4.1. Continuous dependence on parameters.- 4.2. Semiconcavity.- 5. Viscosity Solutions.- 5.1. Definitions.- 5.2. Some properties.- 6. Uniqueness of Viscosity Solutions.- 6.1. A uniqueness theorem.- 6.2. Proofs of Lemmas 6.6 and 6.7.- 7. Historical Remarks.- 5. The Relationship Between the Maximum Principle and Dynamic Programming.- 1. Introduction.- 2. Classical Hamilton-Jacobi Theory.- 3. Relationship for Deterministic Systems.- 3.1. Adjoint variable and value function: Smooth case.- 3.2. Economic interpretation.- 3.3. Methods of characteristics and the Feynman-Kac formula.- 3.4. Adjoint variable and value function: Nonsmooth case.- 3.5. Verification theorems.- 4. Relationship for Stochastic Systems.- 4.1. Smooth case.- 4.2. Nonsmooth case: Differentials in the spatial variable.- 4.3. Nonsmooth case: Differentials in the time variable.- 5. Stochastic Verification Theorems.- 5.1. Smooth case.- 5.2. Nonsmooth case.- 6. Optimal Feedback Controls.- 7. Historical Remarks.- 6. Linear Quadratic Optimal Control Problems.- 1. Introduction.- 2. The Deterministic LQ Problems Revisited.- 2.1. Formulation.- 2.2. A minimization problem of a quadratic functional.- 2.3. A linear Hamiltonian system.- 2.4. The Riccati equation and feedback optimal control.- 3. Formulation of Stochastic LQ Problems.- 3.1. Statement of the problems.- 3.2. Examples.- 4. Finiteness and Solvability.- 5. A Necessary Condition and a Hamiltonian System.- 6. Stochastic Riccati Equations.- 7. Global Solvability of Stochastic Riccati Equations.- 7.1. Existence: The standard case.- 7.2. Existence: The case C = 0, S = 0, and Q, G ?0.- 7.3. Existence: The one-dimensional case.- 8. A Mean-variance Portfolio Selection Problem.- 9. Historical Remarks.- 7. Backward Stochastic Differential Equations.- 1. Introduction.- 2. Linear Backward Stochastic Differential Equations.- 3. Nonlinear Backward Stochastic Differential Equations.- 3.1. BSDEs in finite deterministic durations: Method of contraction mapping.- 3.2. BSDEs in random durations: Method of continuation.- 4. Feynman—Kac-Type Formulae.- 4.1. Representation via SDEs.- 4.2. Representation via BSDEs.- 5. Forward—Backward Stochastic Differential Equations.- 5.1. General formulation and nonsolvability.- 5.2. The four-step scheme, a heuristic derivation.- 5.3. Several solvable classes of FBSDEs.- 6. Option Pricing Problems.- 6.1. European call options and the Black--Scholes formula.- 6.2. Other options.- 7. Historical Remarks.- References.

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Book SynopsisSimple, compact toolkit for designing and analyzing algorithms, with concrete examples from control and communications engineering, artificial intelligence, economic modelling.Trade Review'I highly recommend [this book] to all readers interested in the theory of recursive algorithms and its applications in practice.' Mathematical Reviews'This simple compact toolkit for designing and analyzing stochastic approximation algorithms requires only basic literacy in probability and differential equations … Ideal for graduate students, researchers and practitioners in electrical engineering and computer science, especially those working in control, communications, signal processing and machine learning, this book is also relevant to economics, probability and statistics.' L'Enseignement MathématiqueTable of ContentsPreface; 1. Introduction; 2. Basic convergence analysis; 3. Stability criteria; 4. Lock-in probability; 5. Stochastic recursive inclusions; 6. Multiple timescales; 7. Asynchronous schemes; 8. A limit theorem for fluctuations; 9. Constant stepsize algorithms; 10. Applications; 11. Appendices; References; Index.

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Book SynopsisA textbook for students with some background in probability that develops quickly a rigorous theory of Markov chains and shows how actually to apply it, e.g. to simulation, economics, optimal control, genetics, queues and many other topics, and exercises and examples drawn both from theory and practice.Trade Review'This is an admirable book, treating the topic with mathematical rigour and clarity, mixed with helpful informality; and emphasising numerous applications to a wide range of subjects.' D. V. Lindley, The Mathematical Gazette'My overall impression of this book is very positive … this is the best introduction to the subject that I have come across.' Contemporary Physics'An instructor looking for a suitable text, at the level of a Master of Mathematics degree, can use this book with confidence and enthusiasm.' John Haigh, University of Sussex'We recently based a seminar on this book … it is well suited for an elementary, technically modest, but still rigorous introduction into the heart of a lively and relevant area of stochastic processes.' M. Scheutzow, Zentralblatt MATHTable of ContentsIntroduction; 1. Discrete-time Markov chains; 2. Continuous-time Markov chains I; 3. Continuous-time Markov chains II; 4. Further theory; 5. Applications; Appendix; Probability and measure; Index.

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Book SynopsisThis book gives a comprehensive introduction to numerical methods and analysis of stochastic processes, random fields and stochastic differential equations, and offers graduate students and researchers powerful tools for understanding uncertainty quantification for risk analysis. Coverage includes traditional stochastic ODEs with white noise forcing, strong and weak approximation, and the multi-level Monte Carlo method. Later chapters apply the theory of random fields to the numerical solution of elliptic PDEs with correlated random data, discuss the Monte Carlo method, and introduce stochastic Galerkin finite-element methods. Finally, stochastic parabolic PDEs are developed. Assuming little previous exposure to probability and statistics, theory is developed in tandem with state-of-the-art computational methods through worked examples, exercises, theorems and proofs. The set of MATLAB codes included (and downloadable) allows readers to perform computations themselves and solve the teTrade Review'This book gives both accessible and extensive coverage on stochastic partial differential equations and their numerical solutions. It offers a well-elaborated background needed for solving numerically stochastic PDEs, both parabolic and elliptic. For the numerical solutions it presents not only proofs of convergence results of different numerical methods but also actual implementations, here in Matlab, with technical details included … With numerical implementations hard to find elsewhere in the literature, and a nice presentation of new research findings together with rich references, the book is a welcome companion for anyone working on numerical solutions of stochastic PDEs, and may also be suitable for use in a course on computational stochastic PDEs.' Roger Pettersson, Mathematical ReviewsTable of ContentsPart I. Deterministic Differential Equations: 1. Linear analysis; 2. Galerkin approximation and finite elements; 3. Time-dependent differential equations; Part II. Stochastic Processes and Random Fields: 4. Probability theory; 5. Stochastic processes; 6. Stationary Gaussian processes; 7. Random fields; Part III. Stochastic Differential Equations: 8. Stochastic ordinary differential equations (SODEs); 9. Elliptic PDEs with random data; 10. Semilinear stochastic PDEs.

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Book SynopsisMeyn and Tweedie is back! The bible on Markov chains in general state spaces has been brought up to date to reflect developments in the field since 1996 - many of them sparked by publication of the first edition. The pursuit of more efficient simulation algorithms for complex Markovian models, or algorithms for computation of optimal policies for controlled Markov models, has opened new directions for research on Markov chains. As a result, new applications have emerged across a wide range of topics including optimisation, statistics, and economics. New commentary and an epilogue by Sean Meyn summarise recent developments and references have been fully updated. This second edition reflects the same discipline and style that marked out the original and helped it to become a classic: proofs are rigorous and concise, the range of applications is broad and knowledgeable, and key ideas are accessible to practitioners with limited mathematical background.Trade Review'This second edition remains true to the remarkable standards of scholarship established by the first edition … it will no doubt be a very welcome addition to the literature.' Peter W. Glynn, Prologue to the Second EditionTable of ContentsList of figures; Prologue to the second edition Peter W. Glynn; Preface to the second edition Sean Meyn; Preface to the first edition; Part I. Communication and Regeneration: 1. Heuristics; 2. Markov models; 3. Transition probabilities; 4. Irreducibility; 5. Pseudo-atoms; 6. Topology and continuity; 7. The nonlinear state space model; Part II. Stability Structures: 8. Transience and recurrence; 9. Harris and topological recurrence; 10. The existence of Π; 11. Drift and regularity; 12. Invariance and tightness; Part III. Convergence: 13. Ergodicity; 14. f-Ergodicity and f-regularity; 15. Geometric ergodicity; 16. V-Uniform ergodicity; 17. Sample paths and limit theorems; 18. Positivity; 19. Generalized classification criteria; 20. Epilogue to the second edition; Part IV. Appendices: A. Mud maps; B. Testing for stability; C. Glossary of model assumptions; D. Some mathematical background; Bibliography; Indexes.

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Book SynopsisThis book, first published in 2000, addresses financial mathematics of pricing and hedging derivative securities in uncertain and changing market volatility. The mathematics is introduced through examples and illustrated with simulations, and the modeling approach described is validated and tested on market data. The material is suitable for a one-semester course for graduate students.Trade Review'… provides a good overview to the theoretical and practical problems when dealing with stochastic volatility'. Ralf Korn, Mathematical Methods of Operations Research'… something genuinely new … explained with admirable clarity in this extremely well-written book … [which] is short and to the point, and the production quality is high. Buy it.' Mark Davis, Risk Magazine'… well written and makes ideal reading for a graduate course on mathematical finance. The authors took great care in making their ideas clear. I support this text strongly and recommend it for the intended audience.' P. A. L. Embrechts, Publication of the International Statistical Institute'Thanks to a well-written first chapter on the Black-Scholes theory of derivative pricing, the book is essentially self-contained if one has some basic knowledge in stochastic methods and arbitrage pricing. Its style is largely informal which makes it also accessible to practitioners in the finance industry.' M. Schweizer, Zentralblatt für Mathematik'… an excellent book that succeeds admirably in all its aims. It can satisfy both practitioners and researchers at the same time. It is very well written and it is concise and informative.' Angelos Dassios, The Statistician'I consider this book to be an outstanding achievement. the theory is practically very relevant and scientifically on a high level. The book also serves as a good introduction into the basic ideas of Mathematical Finance, putting emphasis on the techniques of partial differential equations. It can therefore also be recommended to readers with little knowledge about Mathematical Finance.' Monatshefte für MathematikTable of Contents1. The Black-Scholes theory of derivative pricing; 2. Introduction to stochastic volatility models; 3. Scales in mean-reverting stochastic volatility; 4. Tools for estimating the rate of mean-reversion; 5. Symptotics for pricing European derivatives; 6. Implementation and stability; 7. Hedging strategies; 8. Application to exotic derivatives; 9. Application to American derivatives; 10. Generalizations; 11. Applications to interest rates models.

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Book SynopsisThis book provides probabilists with sufficient background to begin applying PDEs to probability theory and probability theory to PDEs. It covers the theory of linear and second order PDEs of parabolic and elliptic type. While most of the techniques described have antecedents in probability theory, the book does cover a few purely analytic techniques.Trade Review'The book will capture your attention with elegant proofs presented in an almost perfectly self-contained manner, with abundant talk in a lecturer's tone by the author himself, but with a little bit of an aficionado's taste. The book, arranged idiosyncratically, has such a strong impact that, at the next moment, you may find yourself carried away in looking for mathematical treasures scattered here and there in each chapter. The reviewer recommends the present book with confidence to anyone who in interested in PDE and probability theory. At least you should always keep this at your side if you are a probabilist at all.' Isamu Doku, Mathematical ReviewsTable of Contents1. Kolmogorov's forward, basic results; 2. Non-elliptic regularity results; 3. Preliminary elliptic regularity results; 4. Nash theory; 5. Localization; 6. On a manifold; 7. Subelliptic estimates and Hörmander's theorem.

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