Stochastics Books

Book SynopsisSince the first edition of Stochastic Modelling for Systems Biology, there have been many interesting developments in the use of likelihood-free methods of Bayesian inference for complex stochastic models. Having been thoroughly updated to reflect this, this third edition covers everything necessary for a good appreciation of stochastic kinetic modelling of biological networks in the systems biology context. New methods and applications are included in the book, and the use of R for practical illustration of the algorithms has been greatly extended. There is a brand new chapter on spatially extended systems, and the statistical inference chapter has also been extended with new methods, including approximate Bayesian computation (ABC). Stochastic Modelling for Systems Biology, Third Edition is now supplemented by an additional software library, written in Scala, described in a new appendix to the book.New in the Third Edition New chapter on sTrade Review"...stochastic modeling has drawn the attention of many researchers in biology and physiology. A textbook, with much elaboration, is highly valuable to understanding the underlying mathematical and computational methods in biological stochastic modeling. Prof Wilkinson has designed the content of this book to fill a gap in the educational text/reference books available for students/researchers learning about stochastic modeling in biological systems... This third edition book almost covers all of the material necessary for students studying stochastic kinetics modelling. The exercises in every chapter certainly illustrate the theory and concept of the book. Appendices A and B elaborate on all of the SBML code and other software associated with the book. The codes are also complemented by links to the author’s webpage and a GitHub repository. The author must be appreciated for adding so many references for further reading. The content of the book is designed for a one-semester graduate-level course in stochastic modeling in biology. Thus, this book is targeted at master and graduate students in interdisciplinary subjects such as applied mathematics, computational biology, bioinformatics, biophysics, Biochemistry, and biomedical engineering."- Chitaranjan Mahapatra, Appeared in ISCB News, January 2020 Table of ContentsIntroduction to biological modelling Representation of biochemical networks Probability models Stochastic simulation Markov processes Chemical and biochemical kinetics Case studies Beyond the Gillespie algorithm Spatially extended systems Bayesian inference and MCMC Inference for stochastic kinetic models Conclusions Appendices

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Book SynopsisThe book addresses the system performance with a focus on the network-enhanced complexities and developing the engineering-oriented design framework of controllers and filters with potential applications in system sciences, control engineering and signal processing areas. Therefore, it provides a unified treatment on the analysis and synthesis for discrete-time stochastic systems with guarantee of certain performances against network-enhanced complexities with applications in sensor networks and mobile robotics. Such a result will be of great importance in the development of novel control and filtering theories including industrial impact.Key Features Provides original methodologies and emerging concepts to deal with latest issues in the control and filtering with an emphasis on a variety of network-enhanced complexities Gives results of stochastic control and filtering distributed control and filtering, and security control of complex nTable of Contents1 Introduction. 2 Finite-Horizon H∞ Control with Randomly Occurring Non-linearities and Fading Measurements. 3. Finite-Horizon H∞ Consensus Control for Multi-Agent Systems with Missing Measurements. 4 Finite-Horizon Distributed H∞ State Estimation with Stochastic Parameters through Sensor Networks. 5 Finite-Horizon Dissipative Control for State-Saturated Discrete Time-Varying Systems with Missing Measurements. 6 Finite-Horizon H∞ Filtering for State-Saturated Discrete Time-Varying Systems with Packet Dropouts. 7 Finite-Horizon Envelope-Constrained H∞ Filtering with Fading Measurements. 8 Distributed Filtering under Uniform Quantizations and Deception Attacks through Sensor Networks. 9 Event-Triggered Distributed H∞ State Estimation with Packet Dropouts through Sensor Networks. 10 Event-Triggered Consensus Control for Multi-Agent Systems in the Framework of Input-to-State Stability in Probability. 11 Event-Triggered Security Control for Discrete-Time Stochastic Systems subject to Cyber-Attacks. 12 Event-Triggered Consensus Control for Multi-Agent Systems subject to Cyber-Attacks in the Framework of Observers.

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Book SynopsisStochastic differential equations are differential equations whose solutions are stochastic processes. They exhibit appealing mathematical properties that are useful in modeling uncertainties and noisy phenomena in many disciplines. This book is motivated by applications of stochastic differential equations in target tracking and medical technology and, in particular, their use in methodologies such as filtering, smoothing, parameter estimation, and machine learning. It builds an intuitive hands-on understanding of what stochastic differential equations are all about, but also covers the essentials of Itô calculus, the central theorems in the field, and such approximation schemes as stochastic RungeKutta. Greater emphasis is given to solution methods than to analysis of theoretical properties of the equations. The book''s practical approach assumes only prior understanding of ordinary differential equations. The numerous worked examples and end-of-chapter exercises include application-Trade Review'Stochastic differential equations have long been used by physicists and engineers, especially in filtering and prediction theory, and more recently have found increasing application in the life sciences, finance and an ever-increasing range of fields. The authors provide intended users with an intuitive, readable introduction and overview without going into technical mathematical details from the often-demanding theory of stochastic analysis, yet clearly pointing out the pitfalls that may arise if its distinctive differences are disregarded. A large part of the book deals with underlying ideas and methods, such as analytical, approximative and computational, which are illustrated through many insightful examples. Linear systems, especially with additive noise and Gaussian solutions, are emphasized, though nonlinear systems are not neglected, and a large number of useful results and formulas are given. The latter part of the book provides an up to date survey and comparison of filtering and parameter estimation methods with many representative algorithms, and culminates with their application to machine learning.' Peter Kloeden, Johann Wolfgang Goethe-Universität Frankfurt am Main'Overall, this is a very well-written and excellent introductory monograph to SDEs, covering all important analytical properties of SDEs, and giving an in-depth discussion of applied methods useful in solving various real-life problems.' Igor Cialenco, MathSciNet'Chapters are rich in examples, numerical simulations, illustrations, derivations and computational assignment' Martin Ondreját, the European Mathematical Society and the Heidelberg Academy of Sciences and HumanitiesTable of Contents1. Introduction; 2. Some background on ordinary differential equations; 3. Pragmatic introduction to stochastic differential equations; 4. Ito calculus and stochastic differential equations; 5. Probability distributions and statistics of SDEs; 6. Statistics of linear stochastic differential equations; 7. Useful theorems and formulas for SDEs; 8. Numerical simulation of SDEs; 9. Approximation of nonlinear SDEs; 10. Filtering and smoothing theory; 11. Parameter estimation in SDE models; 12. Stochastic differential equations in machine learning; 13. Epilogue.

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Book SynopsisThis textbook presents the design and analysis of experiments that comprise the aspects of classical theory for continuous response, modern procedures for categorical response, and especially for correlated categorical response.Trade ReviewFrom the reviews of the second edition:"[This book] is a useful reference or graduate text to complement more common choices for introductory design of experiment books … the methods are logically and thoroughly developed in a rigorous, yet understandable manner. The emphasis on pharmaceutical applications throughout the book is helpful, because this continues to emerge as an important area of applications. The book would be helpful for statisticians and researchers in pharmaceutical areas once they had gained a solid understanding of the fundamentals of design of experiments." –Journal of the American Statistical Association"The second edition of this book … has been reorganized with a list of topics similar to that of the first edition, but with a revised presentation and order. … much greater emphasis now placed on the analysis aspect of design of experiments. … a useful reference book or graduate text … . The methods are logically and thoroughly developed in a rigorous, yet understandable manner. … The book would be helpful for statisticians and researchers in pharmaceutical areas … ." (Christine M. Anderson Cook, Journal of the American Statistical Association, Vol. 98 (463), 2003)"This book is mostly concerned with the mathematical detail of the topics in the contents. There are a few sets of data, to illustrate the material; on these, SAS, S-PLUS or SPSS is used for analysis. … This would be an excellent book for mathematics students who take a course in statistics, or graduate statistic students … ." (N. R. Draper, Short Book Reviews, Vol. 23 (1), 2003)"Helge Toutenburg describes this text as a ‘resource/reference book which contains statistical methods used by researchers in applied areas.’ … the theory is described in a shorthand style that gets to the point without overburdening the reader with mathematical detail … . the author includes thorough discussions of generalized linear models (categorical data analysis) and repeated-measures designs. … a useful, self-contained reference for those who want a quick description of the underlying theory and practice for a large assortment of standard DOE problems." (Peter Wludyka, Technometrics, Vol. 45 (2), May, 2003)From the reviews of the third edition:“This book provides matter related to experimental designs which are of practical relevance. One can understand the subject matter without knowledge of high level mathematics. The book is suitable as a textbook for courses on experimental design in universities and institutions and as a resource book for researchers.” (B. L. Agarwal, Zentralblatt MATH, Vol. 1211, 2011)Table of ContentsComparison of Two Samples.- The Linear Regression Model.- Single#x2013;Factor Experiments with Fixed and Random Effects.- More Restrictive Designs.- Incomplete Block Designs.- Multifactor Experiments.- Models for Categorical Response Variables.- Repeated Measures Model.- Cross#x2013;Over Design.- Statistical Analysis of Incomplete Data.

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Book SynopsisBasic R Programming.- Random Variable Generation.- Monte Carlo Integration.- Controlling and Accelerating Convergence.- Monte Carlo Optimization.- Metropolis#x2013;Hastings Algorithms.- Gibbs Samplers.- Convergence Monitoring and Adaptation for MCMC Algorithms.Trade ReviewFrom the reviews:“Robert and Casella’s new book uses the programming language R, a favorite amongst (Bayesian) statisticians to introduce in eight chapters both basic and advanced Monte Carlo techniques … . The book could be used as the basic textbook for a semester long course on computational statistics with emphasis on Monte Carlo tools … . useful for (and should be next to the computer of) a large body of hands on graduate students, researchers, instructors and practitioners … .” (Hedibert Freitas Lopes, Journal of the American Statistical Association, Vol. 106 (493), March, 2011)“Chapters focuses on MCMC methods the Metropolis–Hastings algorithm, Gibbs sampling, and monitoring and adaptation for MCMC algorithms. … There are exercises within and at the end of all chapters … . Overall, the level of the book makes it suitable for graduate students and researchers. Others who wish to implement Monte Carlo methods, particularly MCMC methods for Bayesian analysis will also find it useful.” (David Scott, International Statistical Review, Vol. 78 (3), 2010)“The primary audience is graduate students in statistics, biostatistics, engineering, etc. who need to know how to utilize Monte Carlo simulation methods to analyze their experiments and/or datasets. … this text does an effective job of including a selection of Monte Carlo methods and their application to a broad array of simulation problems. … Anyone who is an avid R user and has need to integrate and/or optimize complex functions will find this text to be a necessary addition to his or her personal library.” (Dean V. Neubauer, Technometrics, Vol. 53 (2), May, 2011)Table of ContentsBasic R Programming.- Random Variable Generation.- Monte Carlo Integration.- Controlling and Accelerating Convergence.- Monte Carlo Optimization.- Metropolis#x2013;Hastings Algorithms.- Gibbs Samplers.- Convergence Monitoring and Adaptation for MCMC Algorithms.

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Book SynopsisAt the same time it shows how functional data can be studied through parameter-free statistical ideas, and offers an original presentation of new nonparametric statistical methods for functional data analysis.Trade ReviewFrom the reviews: "This is certainly a very valuable book for anyone interested in this new methodology." N.D.C. Veraverbeke for Short Book Reviews of the ISI, December 2006 "The present book does bring something new and, indeed some novel theoretical investigations into the kinds of functional data problems … . I do think the present book is a worthy contribution to the literature. The authors have done a nice job of summarizing some of ongoing research … . Researchers in the growing functional statistics community should be glad to have a copy of the book." (Z. Q. John Lu, Technometrics, Vol. 49 (2), 2007) "This book presents new nonparametric staustical methods for samples of functional data … . The computational aspects of the book are oriented toward practitioners whereas open problems emerging from this new field of statistics will attract Ph. D. students and academic researchers. This book is also accessible to graduate students starting out in the area of functional statistics." (Fazil A. Aliev, Mathematical Reviews, Issue 2007 b) "Nonparametric Functional Data Analysis explores nonparametric methods as that can be applied to functional data, developing new methods and providing theoretical results for the conditional and unconditional mean, median, and mode for independent and dependent functional data. … As a resource for those interested in FDA research and methods, it is highly recommended. … This book should spur new and exciting research in FDA, and it provides new tools that are ready for application to real data sets." (Mark Greenwood, Journal of the American Statistical Association, Vol. 102 (479), 2007) "Example data sets that motivate the development of the models are also provided. … The index provided seems to be fairly complete and is helpful in looking up topics discusses in this monograph. Several chapters end in a section in which the authors provide additional comments, discussions and pose some open problems in this area, which should be appealing for researchers in this field. … This book should be useful for all people interested in the area of functional data analysis." (Anatolij Dvurecenskij, Zentralblatt MATH, Vol. 1119 (21), 2007)Table of ContentsIntroduction to functional nonparametric statistics.- Some functional datasets and associated statistical problematics.- What is a well adapted space for functional data?.- Local weighting of functional variables.- Functional nonparametric prediction methodologies.- Some selected asymptotics.- Computational issues.- Nonparametric supervised classification for functional data.- Nonparametric unsupervised classification for functional data.- Mixing, nonparametric and functional statistics.- Some selected asymptotics.- Application to continuous time processes prediction.- Small ball probabilities, semi-metric spaces and nonparametric statistics.- Conclusion and perspectives.

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Book Synopsis1 Set.- 1.1 Sample sets.- 1.2 Operations with sets.- 1.3 Various relations.- 1.4 Indicator.- Exercises.- 2 Probability.- 2.1 Examples of probability.- 2.2 Definition and illustrations.- 2.3 Deductions from the axioms.- 2.4 Independent events.- 2.5 Arithmetical density.- Exercises.- 3 Counting.- 3.1 Fundamental rule.- 3.2 Diverse ways of sampling.- 3.3 Allocation models; binomial coefficients.- 3.4 How to solve it.- Exercises.- 4 Random Variables.- 4.1 What is a random variable?.- 4.2 How do random variables come about?.- 4.3 Distribution and expectation.- 4.4 Integer-valued random variables.- 4.5 Random variables with densities.- 4.6 General case.- Exercises.- Appendix 1: Borel Fields and General Random Variables.- 5 Conditioning and Independence.- 5.1 Examples of conditioning.- 5.2 Basic formulas.- 5.3 Sequential sampling.- 5.4 Pólya's urn scheme.- 5.5 Independence and relevance.- 5.6 Genetical models.- Exercises.- 6 Mean, Variance, and Transforms.- 6.1 Basic properties of expectationTrade Review"In spite of the original edition of the book being nearly thirty years old, the text still has its role to play in first and second year undergraduate probability courses. It provides an excellent foundation to more advanced courses in the subject."Short Book Reviews, Vol. 23/3, Dec. 2003 "This edition is the third revision of a text on mathematical probability first published in 1974. The text is aimed at undergraduate mathematics students and is accessible to a general audience. The prose is accurate, entertaining, and dense with historical tidbits. Two concluding chapters on mathematical finance have been added to the eight chapters in the third edition by the second author." The American Statistician, May 2004 From the reviews of the fourth edition: "The main novelty in the fourth edition of this well-written book is the addition of new chapters … . The new chapters share the friendly yet rigorous style of the former ones. They begin with an account of the financial vocabulary, which is then expounded in probabilistic terms. … Almost thirty years after its first edition, this charming book continues to be an excellent text for teaching and for self study." (Ricardo Maronna, Statistical Papers, Vol. 45 (4), 2004)Table of ContentsSet * Probability * Counting * Random Variables * Conditioning and Independence * Mean, Variance and Transforms * Poisson and Normal Distributions * From Random Walks to Markov Chains * Mean-Variance Pricing Model * Option Pricing Theory

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Book Synopsis1 Weak convergence of stochastic processes.- 1.1 Introductory remarks.- 1.2 Weak convergence in Rm.- 1.3 Weak convergence in metric spaces.- 1.4 The space D of càdlàg functions.- 1.5 J-continuous functionals.- 1.6 J-convergence of càdlàg processes.- 2 Weak convergence of randomly stopped stochastic processes.- 2.1 Introductory remarks.- 2.2 Randomly stopped scalar càdlàg processes.- 2.3 Randomly stopped vector càdlàg processes.- 2.4 Weakened continuity conditions.- 2.5 Iterated weak limits.- 2.6 Scalar compositions of càdlàg processes.- 2.7 Vector compositions of càdlàg processes.- 2.8 Translation theorems.- 2.9 Randomly stopped locally compact càdlàg processes.- 3 J-convergence of compositions of stochastic processes.- 3.1 Introductory remarks.- 3.2 Compositions with asymptotically continuous components.- 3.3 Asymptotically continuous external processes.- 3.4 Asymptotically continuous internal stopping processes.- 3.5 Semi-vector compositions of càdlàg functions.- 3.6 Semi-vector compositions of càdlàg processes.- 3.7 Vector compositions of càdlàg functions.- 3.8 Vector compositions of càdlàg processes.- 4 Summary of applications.- 4.1 Introductory remarks.- 4.2 Randomly stopped sum-processes.- 4.3 Generalised exceeding processes.- 4.4 Step generalised exceeding processes.- 4.5 Sum-processes with renewal stopping.- 4.6 Accumulation processes.- 4.7 Extremes with random sample size.- 4.8 Mixed sum-max processes.- 4.9 Max-processes with renewal stopping.- 4.10 Shock processes.- Bibliographical remarks.- References.Table of Contents1 Weak convergence of stochastic processes.- 1.1 Introductory remarks.- 1.2 Weak convergence in Rm.- 1.3 Weak convergence in metric spaces.- 1.4 The space D of càdlàg functions.- 1.5 J-continuous functionals.- 1.6 J-convergence of càdlàg processes.- 2 Weak convergence of randomly stopped stochastic processes.- 2.1 Introductory remarks.- 2.2 Randomly stopped scalar càdlàg processes.- 2.3 Randomly stopped vector càdlàg processes.- 2.4 Weakened continuity conditions.- 2.5 Iterated weak limits.- 2.6 Scalar compositions of càdlàg processes.- 2.7 Vector compositions of càdlàg processes.- 2.8 Translation theorems.- 2.9 Randomly stopped locally compact càdlàg processes.- 3 J-convergence of compositions of stochastic processes.- 3.1 Introductory remarks.- 3.2 Compositions with asymptotically continuous components.- 3.3 Asymptotically continuous external processes.- 3.4 Asymptotically continuous internal stopping processes.- 3.5 Semi-vector compositions of càdlàg functions.- 3.6 Semi-vector compositions of càdlàg processes.- 3.7 Vector compositions of càdlàg functions.- 3.8 Vector compositions of càdlàg processes.- 4 Summary of applications.- 4.1 Introductory remarks.- 4.2 Randomly stopped sum-processes.- 4.3 Generalised exceeding processes.- 4.4 Step generalised exceeding processes.- 4.5 Sum-processes with renewal stopping.- 4.6 Accumulation processes.- 4.7 Extremes with random sample size.- 4.8 Mixed sum-max processes.- 4.9 Max-processes with renewal stopping.- 4.10 Shock processes.- Bibliographical remarks.- References.

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Book Synopsis1 Basic Concepts and Elementary Models.- 1. The Vocabulary of Probability Theory.- 2. Events and Probability.- 3. Random Variables and Their Distributions.- 4. Conditional Probability and Independence.- 5. Solving Elementary Problems.- 6. Counting and Probability.- 7. Concrete Probability Spaces.- Illustration 1. A Simple Model in Genetics: Mendel's Law and HardyWeinberg's Theorem.- Illustration 2. The Art of Counting: The Ballot Problem and the Reflection Principle.- Illustration 3. Bertrand's Paradox.- 2 Discrete Probability.- 1. Discrete Random Elements.- 2. Variance and Chebyshev's Inequality.- 3. Generating Functions.- Illustration 4. An Introduction to Population Theory: GaltonWatson's Branching Process.- Illustration 5. Shannon's Source Coding Theorem: An Introduction to Information Theory.- 3 Probability Densities.- I. Expectation of Random Variables with a Density.- 2. Expectation of Functionals of Random Vectors.- 3. Independence.- 4. Random Variables That Are Not Discrete anTable of Contents1 Basic Concepts and Elementary Models.- 1. The Vocabulary of Probability Theory.- 2. Events and Probability.- 2.1. Probability Space.- 2.2. Two Elementary Probabilistic Models.- 3. Random Variables and Their Distributions.- 3.1. Random Variables.- 3.2. Cumulative Distribution Function.- 4. Conditional Probability and Independence.- 4.1. Independence of Events.- 4.2. Independence of Random Variables.- 5. Solving Elementary Problems.- 5.1. More Formulas.- 5.2. A Small Bestiary of Exercises.- 6. Counting and Probability.- 7. Concrete Probability Spaces.- Illustration 1. A Simple Model in Genetics: Mendel’s Law and Hardy—Weinberg’s Theorem.- Illustration 2. The Art of Counting: The Ballot Problem and the Reflection Principle.- Illustration 3. Bertrand’s Paradox.- 2 Discrete Probability.- 1. Discrete Random Elements.- 1.1. Discrete Probability Distributions.- 1.2. Expectation.- 1.3. Independence.- 2. Variance and Chebyshev’s Inequality.- 2.1. Mean and Variance.- 2.2. Chebyshev’s Inequality.- 3. Generating Functions.- 3.1. Definition and Basic Properties.- 3.2. Independence and Product of Generating Functions.- Illustration 4. An Introduction to Population Theory: Galton—Watson’s Branching Process.- Illustration 5. Shannon’s Source Coding Theorem: An Introduction to Information Theory.- 3 Probability Densities.- I. Expectation of Random Variables with a Density.- 1.1. Univariate Probability Densities.- 1.2. Mean and Variance.- 1.3. Chebyshev’s Inequality.- 1.4. Characteristic Function of a Random Variable.- 2. Expectation of Functionals of Random Vectors.- 2.1. Multivariate Probability Densities.- 2.2. Covariance, Cross-Covariance, and Correlation.- 2.3. Characteristic Function of a Random Vector.- 3. Independence.- 3.1. Independent Random Variables.- 3.2. Independent Random Vectors.- 4. Random Variables That Are Not Discrete and Do Not Have a pd.- 4.1. The Abstract Definition of Expectation.- 4.2. Lebesgue’s Theorems and Applications.- Illustration 6. Buffon’s Needle: A Problem in Random Geometry.- 4 Gauss and Poisson.- 1. Smooth Change of Variables.- 1.1. The Method of the Dummy Function.- 1.2. Examples.- 2. Gaussian Vectors.- 2.1. Characteristic Function of Gaussian Vectors.- 2.2. Probability Density of a Nondegenerate Gaussian Vector.- 2.3. Moments of a Centered Gaussian Vector.- 2.4. Random Variables Related to Gaussian Vectors.- 3. Poisson Processes.- 3.1. Homogeneous Poisson Processes Over the Positive Half Line.- 3.2. Nonhomogeneous Poisson Processes Over the Positive Half Line.- 3.3. Homogeneous Poisson Processes on the Plane.- 4. Gaussian Stochastic Processes.- 4.1. Stochastic Processes and Their Law.- 4.2. Gaussian Stochastic Processes.- Illustration 7. An Introduction to Bayesian Decision Theory: Tests of Gaussian Hypotheses.- 5 Convergences.- 1. Almost-Sure Convergence.- 1.1. The Borel—Cantelli Lemma.- 1.2. A Criterion for Almost-Sure Convergence.- 1.3. The Strong Law of Large Numbers.- 2. Convergence in Law.- 2.1. Criterion of the Characteristic Function.- 2.2. The Central Limit Theorem.- 3. The Hierarchy of Convergences.- 3.1. Almost-Sure Convergence Versus Convergence in Probability.- 3.2. Convergence in the Quadratic Mean.- 3.3. Convergence in Law in the Hierarchy of Convergences.- 3.4. The Hierarchical Tableau.- Illustration 8. A Statistical Procedure: The Chi-Square Test.- Illustration 9. Introduction to Signal Theory: Filtering.- Additional Exercises.- Solutions to Additional Exercises.

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Book Synopsis[1] The Taming of Chance.- From Unpredictable to Lawful.- Probability.- Order in the Large.- The Normal Law.- Is It Random?.- More About the Law of Large Numbers.- Where We Stand Now.- [2] Uncertainty and Information.- Messages and Information.- Entropy.- Messages, Codes, and Entropy.- Approximate Entropy.- Again, Is It Random?.- The Perception of Randomness.- [3] Janus-Faced Randomness.- Is Determinism an Illusion?.- Generating Randomness.- Janus and the Demons.- [4] Algorithms, Information, and Chance.- Algorithmic Randomness.- Algorithmic Complexity and Undecidability.- Algorithmic Probability.- [5] The Edge of Randomness.- Between Order and Disorder.- Self-Similarity and Complexity.- What Good is Randomness?.- Sources and Further Readings.- Technical Notes.- Appendix A: Geometric Sums.- Appendix B: Binary Numbers.- Appendix C: Logarithims.- References.Trade ReviewFrom the reviews: THE AMERICAN STATISTICIAN "In summary, I think that many readers with a strong interest in mathematics, statistics, physics, or other areas of science will find this book interesting and challenging. I strongly recommend it to all who are interested in science and would like to see how the ideas of both theoretical mathematics and statistics have been observed and used in real life throughout history." MATHEMATICAL REVIEWS "The book is nicely written and should entertain many readers…"Table of Contents[1] The Taming of Chance.- From Unpredictable to Lawful.- Probability.- Order in the Large.- The Normal Law.- Is It Random?.- More About the Law of Large Numbers.- Where We Stand Now.- [2] Uncertainty and Information.- Messages and Information.- Entropy.- Messages, Codes, and Entropy.- Approximate Entropy.- Again, Is It Random?.- The Perception of Randomness.- [3] Janus-Faced Randomness.- Is Determinism an Illusion?.- Generating Randomness.- Janus and the Demons.- [4] Algorithms, Information, and Chance.- Algorithmic Randomness.- Algorithmic Complexity and Undecidability.- Algorithmic Probability.- [5] The Edge of Randomness.- Between Order and Disorder.- Self-Similarity and Complexity.- What Good is Randomness?.- Sources and Further Readings.- Technical Notes.- Appendix A: Geometric Sums.- Appendix B: Binary Numbers.- Appendix C: Logarithims.- References.

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Book SynopsisContent.- 1: Probability Models.- 1.1 Background.- 1.2 Exchangeability.- 1.4 DeFinetti's Representation Theorem.- 1.5 Proofs of DeFinetti's Theorem and Related Results*.- 1.6 Infinite-Dimensional Parameters*.- 1.7 Problems.- 2: Sufficient Statistics.- 2.1 Definitions.- 2.2 Exponential Families of Distributions.- 2.4 Extremal Families*.- 2.5 Problems.- Chapte 3: Decision Theory.- 3.1 Decision Problems.- 3.2 Classical Decision Theory.- 3.3 Axiomatic Derivation of Decision Theory*.- 3.4 Problems.- 4: Hypothesis Testing.- 4.1 Introduction.- 4.2 Bayesian Solutions.- 4.3 Most Powerful Tests.- 4.4 Unbiased Tests.- 4.5 Nuisance Parameters.- 4.6 P-Values.- 4.7 Problems.- 5: Estimation.- 5.1 Point Estimation.- 5.2 Set Estimation.- 5.3 The Bootstrap*.- 5.4 Problems.- 6: Equivariance*.- 6.1 Common Examples.- 6.2 Equivariant Decision Theory.- 6.3 Testing and Confidence Intervals*.- 6.4 Problems.- 7: Large Sample Theory.- 7.1 Convergence Concepts.- 7.2 Sample Quantiles.- 7.3 Large Sample Estimation.- 7.4 Large Sample Properties of Posterior Distributions.- 7.5 Large Sample Tests.- 7.6 Problems.- 8: Hierarchical Models.- 8.1 Introduction.- 8.3 Nonnormal Models*.- 8.4 Empirical Bayes Analysis*.- 8.5 Successive Substitution Sampling.- 8.6 Mixtures of Models.- 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.1.1Mathematical 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.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 SynopsisThis study discusses the history of the central limit theorem and related probabilistic limit theorems from about 1810 through 1950.Trade ReviewFrom the book reviews:“Fischer provides thorough mathematical descriptions of the development of the central limit theorem as it evolves with increasing mathematical rigor. … Fischer has probably written what will be the definitive history of the central limit theorem for many years to come. … Fischer overflows with detail, insight and excellent commentary.” (David Bellhouse, Historia Mathematica, Vol. 39, 2012)“The book will be of interest not only to professionals in the area of probability and statistics but to a wider audience. … The author has been using a huge amount of sources and archives, including his own works, and he is successful in his goal to describe a comprehensive picture of the development of the CLT. … the book would be an excellent source for student projects on topics from probability and its applications.” (Jordan M. Stoyanov, Zentralblatt MATH, Vol. 1226, 2012)“This work details the history of the central limit theorem and related probabilistic limit theorems roughly from 1810 through 1950, but focuses on 1810 to 1935. … Hans Fischer … authors many papers on the history of mathematics. His skill in both these areas allows him to reveal here the historical development of this important theorem in a way that can easy be adapted to the lecture hall or used in independent study.” (Tom Schulte, The Mathematical Association of America, February, 2011)“The history of the CLT deserves a place of its own, and this book by Hans Fischer is the best … in tracing its development in meticulous historical detail and with mathematical precision. … The book by Hans Fischer is highly recommended as a well-researched comprehensive history of the CLT. One finds here the story of a galaxy of brilliant mathematicians … engaging in the quest for, and debates on, the true meaning and the correct derivation of a beautiful intriguing result.”­­­ (Rabi Bhattacharya, SIAM Review, Vol. 53 (4), 2011)Table of ContentsPreface.- Introduction.- The central limit theorem from laplace to cauchy: changes in stochastic objectives and in analytical methods.- The hypothesis of elementary errors.- Chebyshev's and markov's contributions.- The way towards modern probability.- General limit problems.- Conclusion: the central limit theorem as a link between classical and modern probability.- Index.- Bibliography

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

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

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

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

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

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

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

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

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Book SynopsisTrade Review“This book is an excellent survey, respectively introduction, into recent developments in free probability theory and its applications to random matrices. The authors superbly guide the reader through a number of important examples and present a carefully selected list of 207 relevant publications.” (Ludwig Paditz, zbMATH 1387.60005, 2018)Table of Contents1. Asymptotic Freeness of Gaussian Random Matrices.- 2. The Free Central Limit Theorem and Free Cumulants.- 3. Free Harmonic Analysis.- 4. Asymptotic Freeness.- 5. Second Order Freeness.- 6. Free Group Factors and Freeness.- 7. Free Entropy X-the Microstates Approach via Large Deviations.- Free Entropy X*-the Non-Microstates Approach via Free Fisher Information.- 9. Operator-Valued Free Probability Theory and Block Random Matrices.- 10. Polynomials in Free Variables and Operator-Valued Convolution.- 11. Brown Measure.- Solutions to Exercises.- References.- Index of Exercises.

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Book SynopsisWe propose results of the investigation of the problem of mean square optimal estimation of linear functionals constructed from unobserved values of stationary stochastic processes. Estimates are based on observations of the processes with additive stationary noise process. The aim of the book is to develop methods for finding the optimal estimates of the functionals in the case where some observations are missing. Formulas for computing values of the mean-square errors and the spectral characteristics of the optimal linear estimates of functionals are derived in the case of spectral certainty, where the spectral densities of the processes are exactly known. The minimax robust method of estimation is applied in the case of spectral uncertainty, where the spectral densities of the processes are not known exactly while some classes of admissible spectral densities are given. The formulas that determine the least favourable spectral densities and the minimax spectral characteristics of the optimal estimates of functionals are proposed for some special classes of admissible densities.

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Book SynopsisThe papers collected here focus on probabilistic causality, addressing topics such as the search for causal mechanisms, epistemic and metaphysical views of causality, Bayesian nets and causal dependence, and causation in the special sciences. Some papers stress the statistical analysis of probabilistic data; others address causal issues in physics, with an emphasis on physical processes that are also probabilistic—i.e., stochastic processes.

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Book SynopsisThe monograph is devoted to an investigation of co-operative effects in stochastic models. It includes original results of the authors in the last decade. The main object of the monograph is an analysis of an influence of a stochastic model structure on its characteristics. Problems of a co-operation and a decomposition are actual in a solution of a lot of concrete problems. These problems are: a parallelisation of algorithms and programs, a modelling of supercomputers, computer networks, systems of mobile telephones catastrophes in complex systems, a design and an improvement of technological and economical processes etc. The co-operative effects create a source of significant dependencies between complex system characteristics under large random disturbances. To analyse these effects is necessary to create special methods based on structural analysis of multi-element stochastic models together with majoral asymptotic bounds of these models characteristics. At the same time it demands to develop new approaches to a processing of statistical data and a skill in an usage of the probability theory limit theorems and related asymptotic series and bounds. A choice of the monograph material is defined as by initial applied problems so by probability methods of their solution. Conditionally the monograph may be divided into two parts. First of them contains four sections devoted to a finding of the co-operative effects and to a development of new related analytical and numerical methods. This part has presumably methodological character and creates a theoretical base of an investigation of applied stochastic systems. Second part contains three sections devoted to a solution of different applied problems. It has some interesting substantial results.

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Book SynopsisUncertainty and risk are integral to engineering because real systems have inherent ambiguities that arise naturally or due to our inability to model complex physics. The authors discuss probability theory, stochastic processes, estimation, and stochastic control strategies and show how probability can be used to model uncertainty in control and estimation problems. The material is practical and rich in research opportunities.The authors provide a comprehensive treatment of stochastic systems from the foundations of probability to stochastic optimal control. The book covers discrete- and continuous-time stochastic dynamic systems leading to the derivation of the Kalman filter, its properties, and its relation to the frequency domain Wiener filter as well as the dynamic programming derivation of the linear quadratic Gaussian (LQG) and the linear exponential Gaussian (LEG) controllers and their relation to H2 and H-inf controllers and system robustness.Stochastic Processes, Estimation, and Control is divided into three related sections. First, the authors present the concepts of probability theory, random variables, and stochastic processes, which lead to the topics of expectation, conditional expectation, and discrete-time estimation and the Kalman filter. After establishing this foundation, stochastic calculus and continuous-time estimation are introduced. Finally, dynamic programming for both discrete-time and continuous-time systems leads to the solution of optimal stochastic control problems, resulting in controllers with significant practical application.

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Book SynopsisThis book is concerned with the theory of stochastic processes and the theoretical aspects of statistics for stochastic processes. It combines classic topics such as construction of stochastic processes, associated filtrations, processes with independent increments, Gaussian processes, martingales, Markov properties, continuity and related properties of trajectories with contemporary subjects: integration with respect to Gaussian processes, Itȏ integration, stochastic analysis, stochastic differential equations, fractional Brownian motion and parameter estimation in diffusion models. Table of ContentsPreface xi Introduction xiii Part 1 Theory of Stochastic Processes 1 Chapter 1 Stochastic Processes General Properties. Trajectories, Finite-dimensional Distributions 3 1.1 Definition of a stochastic process 3 1.2 Trajectories of a stochastic process Some examples of stochastic processes 5 1.2.1 Definition of trajectory and some examples 5 1.2.2 Trajectory of a stochastic process as a random element.8 1.3 Finite-dimensional distributions of stochastic processes: consistency conditions.10 1.3.1 Definition and properties of finite-dimensional distributions 10 1.3.2 Consistency conditions.11 1.3.3 Cylinder sets and generated σ-algebra 13 1.3.4 Kolmogorov theorem on the construction of a stochastic process by the family of probability distributions 15 1.4 Properties of σ-algebra generated by cylinder sets. The notion of σ-algebra generated by a stochastic process 19 Chapter 2 Stochastic Processes with Independent Increments 21 2.1 Existence of processes with independent increments in terms of incremental characteristic functions 21 2.2 Wiener process 24 2.2.1 One-dimensional Wiener process 24 2.2.2 Independent stochastic processes Multidimensional Wiener process 24 2.3 Poisson process 27 2.3.1 Poisson process defined via the existence theorem 27 2.3.2 Poisson process defined via the distributions of the increments 28 2.3.3 Poisson process as a renewal process 30 2.4 Compound Poisson process 33 2.5 Lévy processes 34 2.5.1 Wiener process with a drift 36 2.5.2 Compound Poisson process as a Lévy process 36 2.5.3 Sum of a Wiener process with a drift and a Poisson process 36 2.5.4 Gamma process 37 2.5.5 Stable Lévy motion37 2.5.6 Stable Lévy subordinator with stability parameter α ∈ (0, 1) 38 Chapter 3 Gaussian Processes Integration with Respect to Gaussian Processes 39 3.1 Gaussian vectors 39 3.2 Theorem of Gaussian representation (theorem on normal correlation) 42 3.3 Gaussian processes. 44 3.4 Examples of Gaussian processes 46 3.4.1 Wiener process as an example of a Gaussian process 46 3.4.2 Fractional Brownian motion.48 3.4.3 Sub-fractional and bi-fractional Brownian motion 50 3.4.4 Brownian bridge 50 3.4.5 Ornstein–Uhlenbeck process 51 3.5 Integration of non-random functions with respect to Gaussian processes 52 3.5.1 General approach 52 3.5.2 Integration of non-random functions with respect to the Wiener process 54 3.5.3 Integration w.r.t the fractional Brownian motion 57 3.6 Two-sided Wiener process and fractional Brownian motion: Mandelbrot–van Ness representation of fractional Brownian motion 60 3.7 Representation of fractional Brownian motion as the Wiener integral on the compact integral 63 Chapter 4 Construction, Properties and Some Functionals of the Wiener Process and Fractional Brownian Motion 67 4.1 Construction of a Wiener process on the interval [0, 1] 67 4.2 Construction of a Wiener process on R+ 72 4.3 Nowhere differentiability of the trajectories of a Wiener process 74 4.4 Power variation of the Wiener process and of the fractional Brownian motion77 4.4.1 Ergodic theorem for power variations 77 4.5 Self-similar stochastic processes 79 4.5.1 Definition of self-similarity and some examples 79 4.5.2 Power variations of self-similar processes on finite intervals.80 Chapter 5 Martingales and Related Processes 85 5.1 Notion of stochastic basis with filtration 85 5.2 Notion of (sub-, super-) martingale: elementary properties 86 5.3 Examples of (sub-, super-) martingales 87 5.4 Markov moments and stopping times 90 5.5 Martingales and related processes with discrete time 96 5.5.1 Upcrossings of the interval and existence of the limit of submartingale 96 5.5.2 Examples of martingales having a limit and of uniformly and non-uniformly integrable martingales 102 5.5.3 Lévy convergence theorem 104 5.5.4 Optional stopping 105 5.5.5 Maximal inequalities for (sub-, super-) martingales 108 5.5.6 Doob decomposition for the integrable processes with discrete time 111 5.5.7 Quadratic variation and quadratic characteristics: Burkholder–Davis–Gundy inequalities 113 5.5.8 Change of probability measure and Girsanov theorem for discrete-time processes 116 5.5.9 Strong law of large numbers for martingales with discrete time 120 5.6 Lévy martingale stopped 126 5.7 Martingales with continuous time 127 Chapter 6 Regularity of Trajectories of Stochastic Processes 131 6.1 Continuity in probability and in L2(Ω,F, P) 131 6.2 Modification of stochastic processes: stochastically equivalent and indistinguishable processes 133 6.3 Separable stochastic processes: existence of separable modification 135 6.4 Conditions of D-regularity and absence of the discontinuities of the second kind for stochastic processes 138 6.4.1 Skorokhod conditions of D-regularity in terms of three-dimensional distributions 138 6.4.2 Conditions of absence of the discontinuities of the second kind formulated in terms of conditional probabilities of large increments 144 6.5 Conditions of continuity of trajectories of stochastic processes 148 6.5.1 Kolmogorov conditions of continuity in terms of two-dimensional distributions 148 6.5.2 Hölder continuity of stochastic processes: a sufficient condition 152 6.5.3 Conditions of continuity in terms of conditional probabilities 154 Chapter 7 Markov and Diffusion Processes 157 7.1 Markov property 157 7.2 Examples of Markov processes 163 7.2.1 Discrete-time Markov chain 163 7.2.2 Continuous-time Markov chain 165 7.2.3 Process with independent increments 168 7.3 Semigroup resolvent operator and generator related to the homogeneous Markov process 168 7.3.1 Semigroup related to Markov process 168 7.3.2 Resolvent operator and resolvent equation 169 7.3.3 Generator of a semigroup.171 7.4 Definition and basic properties of diffusion process 175 7.5 Homogeneous diffusion process Wiener process as a diffusion process 179 7.6 Kolmogorov equations for diffusions 181 Chapter 8 Stochastic Integration 187 8.1 Motivation..187 8.2 Definition of Itô integral 189 8.2.1 Itô integral of Wiener process 195 8.3 Continuity of Itô integral 197 8.4 Extended Itô integral 199 8.5 Itô processes and Itô formula 203 8.6 Multivariate stochastic calculus 212 8.7 Maximal inequalities for Itô martingales 215 8.7.1 Strong law of large numbers for Itô local martingales 218 8.8 Lévy martingale characterization of Wiener process 220 8.9 Girsanov theorem 223 8.10 Itô representation 228 Chapter 9 Stochastic Differential Equations.233 9.1 Definition, solvability conditions, examples 233 9.1.1 Existence and uniqueness of solution 234 9.1.2 Some special stochastic differential equations 238 9.2 Properties of solutions to stochastic differential equations 241 9.3 Continuous dependence of solutions on coefficients 245 9.4 Weak solutions to stochastic differential equations. 247 9.5 Solutions to SDEs as diffusion processe 249 9.6 Viability, comparison and positivity of solutions to stochastic differential equations 252 9.6.1 Comparison theorem for one-dimensional projections of stochastic differential equations 257 9.6.2 Non-negativity of solutions to stochastic differential equations 258 9.7 Feynman–Kac formula 258 9.8 Diffusion model of financial markets 260 9.8.1 Admissible portfolios, arbitrage and equivalent martingale measure 263 9.8.2 Contingent claims, pricing and hedging 266 Part 2 Statistics of Stochastic Processes 271 Chapter 10 Parameter Estimation 273 10.1 Drift and diffusion parameter estimation in the linear regression model with discrete time 273 10.1.1 Drift estimation in the linear regression model with discrete time in the case when the initial value is known 274 10.1.2 Drift estimation in the case when the initial value is unknown 277 10.2 Estimation of the diffusion coefficient in a linear regression model with discrete time 277 10.3 Drift and diffusion parameter estimation in the linear model with continuous time and the Wiener noise 278 10.3.1 Drift parameter estimation 279 10.3.2 Diffusion parameter estimation 280 10.4 Parameter estimation in linear models with fractional Brownian motion 281 10.4.1 Estimation of Hurst index 281 10.4.2 Estimation of the diffusion parameter 283 10.5 Drift parameter estimation 284 10.6 Drift parameter estimation in the simplest autoregressive model 285 10.7 Drift parameters estimation in the homogeneous diffusion model 289 Chapter 11 Filtering Problem Kalman-Bucy Filter 293 11.1 General setting 293 11.2 Auxiliary properties of the non-observable process 294 11.3 What is an optimal filter 295 11.4 Representation of an optimal filter via an integral equation with respect to an observable process 296 11.5 Integral Wiener-Hopf equation 299 Appendices 311 Appendix 1 313 Appendix 2 329 Bibliography 363 Index 369

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Book SynopsisBranching processes are stochastic processes which represent the reproduction of particles, such as individuals within a population, and thereby model demographic stochasticity. In branching processes in random environment (BPREs), additional environmental stochasticity is incorporated, meaning that the conditions of reproduction may vary in a random fashion from one generation to the next. This book offers an introduction to the basics of BPREs and then presents the cases of critical and subcritical processes in detail, the latter dividing into weakly, intermediate, and strongly subcritical regimes.Table of Contents1. Branching Processes in Varying Environment. 2. Branching Processes in Random Environment. 3. Large Deviations for BPREs. 4. Properties of Random Walks. 5. Critical BPREs: the Annealed Approach. 6. Critical BPREs: the Quenched Approach. 7. Weakly Subcritical BPREs. 8. Intermediate Subcritical BPREs. 9. Strongly Subcritical BPREs. 10. Multi-type BPREs.

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Book SynopsisMastering chance has, for a long time, been a preoccupation of mathematical research. Today, we possess a predictive approach to the evolution of systems based on the theory of probabilities. Even so, uncovering this subject is sometimes complex, because it necessitates a good knowledge of the underlying mathematics. This book offers an introduction to the processes linked to the fluctuations in chance and the use of numerical methods to approach solutions that are difficult to obtain through an analytical approach. It takes classic examples of inventory and queueing management, and addresses more diverse subjects such as equipment reliability, genetics, population dynamics, physics and even market finance. It is addressed to those at Master�s level, at university, engineering school or management school, but also to an audience of those in continuing education, in order that they may discover the vast field of decision support.Table of ContentsPart 1. Basic Mathematical Concepts 1. Basic Reminders of Probability. 2. Probabilistic Models. 3. Inventory Management. Part 2. Stochastic Processes 4. Markov Chains. 5. Markov Processes. 6. Queueing Systems. 7. Various Applications. Part 3. Simulation 8. Generator Programs. 9. Principles of Simulation. 10. Simulation of Inventory Management. 11. Simulation of a Queueing Process. 12. Optimization and Simulation.

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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 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 SynopsisThis title considers the special of random processes known as semi-Markov processes. These possess the Markov property with respect to any intrinsic Markov time such as the first exit time from an open set or a finite iteration of these times. The class of semi-Markov processes includes strong Markov processes, Lévy and Smith stepped semi-Markov processes, and some other subclasses. Extensive coverage is devoted to non-Markovian semi-Markov processes with continuous trajectories and, in particular, to semi-Markov diffusion processes. Readers looking to enrich their knowledge on Markov processes will find this book a valuable resource.Table of ContentsIntroduction 9 Chapter 1. Stepped Semi-Markov Processes 17 1.1. Random sequence 17 1.2. Markov chain 20 1.3. Two-dimensional Markov chain 25 1.4. Semi-Markov process 29 1.5. Stationary distributions 32 Chapter 2. Sequences of First Exit Times and Regeneration Times 37 2.1. Basic maps 38 2.2. Markov times 42 2.3. Deducing sequences 46 2.4. Correct exit and continuity 55 2.5. Time of regeneration 63 Chapter 3. General Semi-Markov Processes 71 3.1. Definition of a semi-Markov process 72 3.2. Transition function of a SM process 79 3.3. Operators and SM walk 82 3.4. Operators and SM process 91 3.5. Criterion of Markov property for SM processes 103 3.6. Intervals of constancy 110 Chapter 4. Construction of Semi-Markov Processes using Semi-Markov Transition Functions 115 4.1. Realization of an innite systemof pairs 116 4.2. Extension of a measure 121 4.3. Construction of a measure 124 4.4. Construction of a projective system of measures 127 4.5. Semi-Markov processes 133 Chapter 5. Semi-Markov Processes of Diffusion Type 137 5.1. One-dimensional semi-Markov processes of diffusion type 138 5.1.1. Differential equation 138 5.1.2. Construction SM process 143 5.1.3. Some properties of the process 159 5.2. Multi-dimensional processes of diffusion type 168 5.2.1. Differential equations of elliptic type 168 5.2.2. Neighborhood of arbitrary form 171 5.2.3. Neighborhood of spherical form 178 5.2.4. Characteristic operator 188 Chapter 6. Time Change and Semi-Markov Processes 197 6.1. Time change and trajectories 198 6.2. Intrinsic time and traces 206 6.3. Canonical time change 211 6.4. Coordination of function and time change 223 6.5. Random time changes 228 6.6. Additive functionals 233 6.7. Distribution of a time run along the trace 242 6.8. Random curvilinear integrals 252 6.9. Characteristic operator and integral 264 6.10. Stochastic integral 268 6.10.1. Semi-martingale and martingale 268 6.10.2. Stochastic integral 275 6.10.3. Ito-Dynkin’s formula 277 Chapter 7. Limit Theorems for Semi-Markov Processes 281 7.1. Weak compactness and weak convergence 281 7.2. Weak convergence of semi-Markov processes 289 Chapter 8. Representation of a Semi-Markov Process as a Transformed Markov Process 299 8.1. Construction by operator 300 8.2. Comparison of processes 302 8.3. Construction by parameters of Lévy formula 307 8.4. Stationary distribution 311 Chapter 9. Semi-Markov Model of Chromatography 325 9.1. Chromatography 326 9.2. Model of liquid column chromatography 328 9.3. Some monotone Semi-Markov processes 332 9.4. Transfer with diffusion 337 9.5. Transfer with nal absorption 346 Bibliography 361 Index 369

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Book SynopsisSwitching processes, invented by the author in 1977, is the main tool used in the investigation of traffic problems from automotive to telecommunications. The title provides a new approach to low traffic problems based on the analysis of flows of rare events and queuing models. In the case of fast switching, averaging principle and diffusion approximation results are proved and applied to the investigation of transient phenomena for wide classes of overloading queuing networks. The book is devoted to developing the asymptotic theory for the class of switching queuing models which covers models in a Markov or semi-Markov environment, models under the influence of flows of external or internal perturbations, unreliable and hierarchic networks, etc.Table of ContentsPreface 13 Definitions 17 Chapter 1. Switching Stochastic Models 19 1.1. Random processes with discrete component 19 1.1.1.Markov and semi-Markov processes 21 1.1.2. Processes with independent increments and Markov switching 21 1.1.3. Processes with independent increments and semi-Markov switching 23 1.2. Switching processes 24 1.2.1. Definition of switching processes 24 1.2.2. Recurrent processes of semi-Markov type (simple case) 26 1.2.3.RPSMwithMarkov switching 26 1.2.4. General case of RPSM 27 1.2.5. Processes with Markov or semi-Markov switching 27 1.3. Switching stochastic models 28 1.3.1. Sums of random variables 29 1.3.2. Random movements 29 1.3.3. Dynamic systems in a random environment 30 1.3.4. Stochastic differential equations in a random environment 30 1.3.5. Branching processes 31 1.3.6. State-dependent flows 32 1.3.7. Two-level Markov systems with feedback 32 1.4. Bibliography 33 Chapter 2. Switching Queueing Models 37 2.1. Introduction 37 2.2. Queueing systems 38 2.2.1. Markov queueing models 38 2.2.1.1. A state-dependent system MQ/MQ/1/∞ 39 2.2.1.2. Queueing system MM,Q/MM,Q/1/m 40 2.2.1.3. System MQ,B/MQ,B/1/∞ 41 2.2.2.Non-Markov systems 42 2.2.2.1. Semi-Markov system SM/MSM,Q/1 42 2.2.2.2. System MSM,Q/MSM,Q/1/∞ 43 2.2.2.3. System MSM,Q/MSM,Q/1/V 44 2.2.3. Models with dependent arrival flows 45 2.2.4. Polling systems 46 2.2.5. Retrial queueing systems 47 2.3. Queueing networks 48 2.3.1. Markov state-dependent networks 49 2.3.1.1. Markov network (MQ/MQ/m/∞)r 49 2.3.1.2. Markov networks (MQ,B/MQ,B/m/∞)r with batches 50 2.3.2.Non-Markov networks 50 2.3.2.1. State-dependent semi-Markov networks 50 2.3.2.2. Semi-Markov networks with random batches 52 2.3.2.3. Networks with state-dependent input 53 2.4.Bibliography 54 Chapter 3. Processes of Sums of Weakly-dependent Variables 57 3.1. Limit theorems for processes of sums of conditionally independent random variables 57 3.2. Limit theorems for sums with Markov switching 65 3.2.1. Flows of rare events 67 3.2.1.1. Discrete time 67 3.2.1.2. Continuous time 69 3.3. Quasi-ergodic Markov processes 70 3.4. Limit theorems for non-homogenous Markov processes 73 3.4.1. Convergence to Gaussian processes 74 3.4.2. Convergence to processes with independent increments 78 3.5. Bibliography 81 Chapter 4. Averaging Principle and Diffusion Approximation for Switching Processes 83 4.1. Introduction 83 4.2. Averaging principle for switching recurrent sequences 84 4.3. Averaging principle and diffusion approximation for RPSMs 88 4.4. Averaging principle and diffusion approximation for recurrent processes of semi-Markov type (Markov case) 95 4.4.1. Averaging principle and diffusion approximation for SMP 105 4.5. Averaging principle for RPSM with feedback 106 4.6. Averaging principle and diffusion approximation for switching processes 108 4.6.1. Averaging principle and diffusion approximation for processes with semi-Markov switching 112 4.7. Bibliography 113 Chapter 5. Averaging and Diffusion Approximation in Overloaded Switching Queueing Systems and Networks 117 5.1. Introduction 117 5.2. Markov queueing models 120 5.2.1. System MQ,B/MQ,B/1/∞ 121 5.2.2. System MQ/MQ/1/∞ 124 5.2.3. Analysis of the waiting time 129 5.2.4. An output process 131 5.2.5. Time-dependent system MQ,t/MQ,t/1/∞ 132 5.2.6. Asystemwith impatient calls 134 5.3. Non-Markov queueing models 135 5.3.1. System GI/MQ/1/∞ 135 5.3.2. Semi-Markov system SM/MSM,Q/1/∞ 136 5.3.3. System MSM,Q/MSM,Q/1/∞ 138 5.3.4. System SMQ/MSM,Q/1/∞ 139 5.3.5. System GQ/MQ/1/∞ 142 5.3.6. A system with unreliable servers 143 5.3.7. Polling systems 145 5.4. Retrial queueing systems 146 5.4.1. Retrial system MQ/G/1/w.r 147 5.4.2. System M¯ /G¯/1/w.r 150 5.4.3. Retrial system M/M/m/w.r 154 5.5. Queueing networks 159 5.5.1. State-dependent Markov network (MQ/MQ/1/∞)r 159 5.5.2. Markov state-dependent networks with batches 161 5.6. Non-Markov queueing networks 164 5.6.1. A network (MSM,Q/MSM,Q/1/∞)r with semi-Markov switching 164 5.6.2. State-dependent network with recurrent input 169 5.7. Bibliography 172 Chapter 6. Systems in Low Traffic Conditions 175 6.1. Introduction 175 6.2. Analysis of the first exit time from the subset of states 176 6.2.1. Definition of S-set 176 6.2.2. An asymptotic behavior of the first exit time 177 6.2.3. State space forming a monotone structure 180 6.2.4. Exit time as the time of first jump of the process of sums with Markov switching 182 6.3. Markov queueing systems with fast service 183 6.3.1. M/M/s/m systems 183 6.3.1.1. System MM/M/l/m in a Markov environment 185 6.3.2. Semi-Markov queueing systems with fast service 188 6.4. Single-server retrial queueing model 190 6.4.1. Case 1: fast service 191 6.4.1.1. State-dependent case 194 6.4.2. Case 2: fast service and large retrial rate 195 6.4.3. State-dependent model in a Markov environment 197 6.5. Multiserver retrial queueing models 201 6.6. Bibliography 204 Chapter 7. Flows of Rare Events in Low and Heavy Traffic Conditions 207 7.1. Introduction 207 7.2. Flows of rare events in systems with mixing 208 7.3. Asymptotically connected sets (Vn-S-sets) 211 7.3.1. Homogenous case 211 7.3.2. Non-homogenous case 214 7.4. Heavy traffic conditions 215 7.5. Flows of rare events in queueing models 216 7.5.1. Light traffic analysis in models with finite capacity 216 7.5.2. Heavy traffic analysis 218 7.6. Bibliography 219 Chapter 8. Asymptotic Aggregation of State Space 221 8.1. Introduction 221 8.2. Aggregation of finite Markov processes (stationary behavior) 223 8.2.1. Discrete time 223 8.2.2. Hierarchic asymptotic aggregation 225 8.2.3. Continuous time 227 8.3. Convergence of switching processes 228 8.4. Aggregation of states in Markov models 231 8.4.1. Convergence of the aggregated process to a Markov process (finite state space) 232 8.4.2. Convergence of the aggregated process with a general state space 236 8.4.3. Accumulating processes in aggregation scheme 237 8.4.4. MP aggregation in continuous time 238 8.5. Asymptotic behavior of the first exit time from the subset of states (non-homogenous in time case) 240 8.6. Aggregation of states of non-homogenous Markov processes 243 8.7. Averaging principle for RPSM in the asymptotically aggregated Markov environment 246 8.7.1. Switching MP with a finite state space 247 8.7.2. Switching MP with a general state space 250 8.7.3. Averaging principle for accumulating processes in the asymptotically aggregated semi-Markov environment 251 8.8. Diffusion approximation for RPSM in the asymptotically aggregated Markov environment 252 8.9. Aggregation of states in Markov queueing models 255 8.9.1. System MQ/MQ/r/∞ with unreliable servers in heavy traffic 255 8.9.2. System MM,Q/MM,Q/1/∞ in heavy traffic 256 8.10. Aggregation of states in semi-Markov queueing models 258 8.10.1. System SM/MSM,Q/1/∞ 258 8.10.2. System MSM,Q/MSM,Q/1/∞ 259 8.11. Analysis of flows of lost calls 260 8.12. Bibliography 263 Chapter 9. Aggregation in Markov Models with Fast Markov Switching 267 9.1. Introduction 267 9.2. Markov models with fast Markov switching 269 9.2.1.Markov processes with Markov switching 269 9.2.2. Markov queueing systems with Markov type switching 271 9.2.3. Averaging in the fast Markov type environment 272 9.2.4. Approximation of a stationary distribution 274 9.3. Proofs of theorems 275 9.3.1. Proof of Theorem 9.1 275 9.3.2. Proof of Theorem 9.2 277 9.3.3. Proof of Theorem 9.3 279 9.4. Queueing systems with fast Markov type switching 279 9.4.1. System MM,Q/MM,Q/1/N 279 9.4.1.1. Averaging of states of the environment 279 9.4.1.2. The approximation of a stationary distribution 280 9.4.2. Batch system BMM,Q/BMM,Q/1/N 281 9.4.3. System M/M/s/mwith unreliable servers 282 9.4.4. Priority model MQ/MQ/m/s,N 283 9.5. Non-homogenous in time queueing models 285 9.5.1. SystemMM,Q,t/MM,Q,t/s/m with fast switching – averaging of states 286 9.5.2. System MM,Q/MM,Q/s/m with fast switching – aggregation of states 287 9.6. Numerical examples 288 9.7. Bibliography 289 Chapter 10. Aggregation in Markov Models with Fast Semi-Markov Switching 291 10.1. Markov processes with fast semi-Markov switches 292 10.1.1.Averaging of a semi-Markov environment 292 10.1.2. Asymptotic aggregation of a semi-Markov environment 300 10.1.3. Approximation of a stationary distribution 305 10.2. Averaging and aggregation in Markov queueing systems with semi-Markov switching 309 10.2.1.Averaging of states of the environment 309 10.2.2. Asymptotic aggregation of states of the environment 310 10.2.3. The approximation of a stationary distribution 311 10.3. Bibliography 313 Chapter 11. Other Applications of Switching Processes 315 11.1. Self-organization in multicomponent interacting Markov systems 315 11.2. Averaging principle and diffusion approximation for dynamic systems with stochastic perturbations 319 11.2.1. Recurrent perturbations 319 11.2.2. Semi-Markov perturbations 321 11.3. Random movements 324 11.3.1. Ergodic case 324 11.3.2. Case of the asymptotic aggregation of state space 325 11.4. Bibliography 326 Chapter 12. Simulation Examples 329 12.1. Simulation of recurrent sequences 329 12.2. Simulation of recurrent point processes 331 12.3. Simulation ofRPSM 332 12.4. Simulation of state-dependent queueing models 334 12.5. Simulation of the exit time from a subset of states of a Markov chain 337 12.6. Aggregation of states in Markov models 340 Index 343

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Book SynopsisThis book provides a pedagogical examination of the way in which stochastic models are encountered in applied sciences and techniques such as physics, engineering, biology and genetics, economics and social sciences. It covers Markov and semi-Markov models, as well as their particular cases: Poisson, renewal processes, branching processes, Ehrenfest models, genetic models, optimal stopping, reliability, reservoir theory, storage models, and queuing systems. Given this comprehensive treatment of the subject, students and researchers in applied sciences, as well as anyone looking for an introduction to stochastic models, will find this title of invaluable use.Table of ContentsPreface ix Chapter 1. Introduction to Stochastic Processes 1 1.1. Sequences of random variables 1 1.2. The notion of stochastic process 10 1.3. Martingales 13 1.4. Markov chains 17 1.5. State classification 24 1.6. Continuous-time Markov processes 27 1.7. Semi-Markov processes 33 Chapter 2. Simple Stochastic Models 37 2.1. Urn models 37 2.2. Random walks 39 2.3. Brownian motion 44 2.4. Poisson processes 50 2.5. Birth and death processes 59 Chapter 3. Elements of Markov Modeling 61 3.1. Markov models: ideas, history, applications 61 3.2. The discrete-time Ehrenfest model 63 3.3. Markov models in genetics 79 3.4. Markov storage models 110 3.5. Reliability of Markov models 124 Chapter 4. Renewal Models 149 4.1. Fundamental concepts and examples 149 4.2. Waiting times 155 4.3. Modified renewal processes 159 4.4. Replacement models 161 4.5. Renewal reward processes 165 4.6. The risk problem of an insurance company 168 4.7. Counter models 171 4.8. Alternating renewal processes 180 4.9. Superposition of renewal processes 182 4.10. Regenerative processes 186 Chapter 5. Semi-Markov Models 189 5.1. Introduction 189 5.2. Markov renewal processes 190 5.3. First-passage times and state classification 196 5.4. Reliability 200 5.5. Reservoir models 207 5.6. Queues 218 5.7. Digital communication channels 222 Chapter 6. Branching Models 227 6.1. The Bienaymé-Galton-Watson model 227 6.2. Generalizations of the B-G-W model 271 6.3. Continuous-time models 302 Chapter 7. Optimal Stopping Models 315 7.1. The classic optimal stopping problem 315 7.2. Renewal with binary decision 333 Bibliography 343 Notation 367 Index 369

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Book SynopsisStochastic finance and financial engineering have been rapidly expanding fields of science over the past four decades, mainly due to the success of sophisticated quantitative methodologies in helping professionals manage financial risks. In recent years, we have witnessed a tremendous acceleration in research efforts aimed at better comprehending, modeling and hedging this kind of risk. These two volumes aim to provide a foundation course on applied stochastic finance. They are designed for three groups of readers: firstly, students of various backgrounds seeking a core knowledge on the subject of stochastic finance; secondly financial analysts and practitioners in the investment, banking and insurance industries; and finally other professionals who are interested in learning advanced mathematical and stochastic methods, which are basic knowledge in many areas, through finance. Volume 1 starts with the introduction of the basic financial instruments and the fundamental principles of financial modeling and arbitrage valuation of derivatives. Next, we use the discrete-time binomial model to introduce all relevant concepts. The mathematical simplicity of the binomial model also provides us with the opportunity to introduce and discuss in depth concepts such as conditional expectations and martingales in discrete time. However, we do not expand beyond the needs of the stochastic finance framework. Numerous examples, each highlighted and isolated from the text for easy reference and identification, are included. The book concludes with the use of the binomial model to introduce interest rate models and the use of the Markov chain model to introduce credit risk. This volume is designed in such a way that, among other uses, makes it useful as an undergraduate course.Table of ContentsPreface xi Chapter 1. Probability and Random Variables 1 1.1. Introductory notes 1 1.2. Probability space 2 1.3. Conditional probability and independence 8 1.4. Random variables 12 1.5. Expectation and variance of a random variable 24 1.6. Jointly distributed random variables 28 1.7. Moment generating functions 32 1.8. Probability inequalities and limit theorems 37 1.9.Multivariate normal distribution 44 Chapter 2. An Introduction to Financial Instruments and Derivatives 49 2.1. Introduction 49 2.2. Bonds and basic interest rates 50 2.3. Forward contracts 58 2.4. Futures contracts 60 2.5.Swaps 60 2.6.Options 62 2.7. Types of market participants 67 2.8.Arbitrage relationships between call and put options 67 2.9.Exercises 69 Chapter 3. Conditional Expectation and Markov Chains 71 3.1. Introduction 71 3.2. Conditional expectation: the discrete case 72 3.3. Applications of conditional expectations 75 3.4. Properties of the conditional expectation 81 3.5. Markov chains 85 3.6. Exercises 131 4.1. Introductory notes 137 Chapter 4. The No-Arbitrage Binomial Pricing Model 137 4.2. Binomial model 138 4.3. Stochastic evolution of the asset prices 141 4.4. Binomial approximation to the lognormal distribution 143 4.5. One-period European call option 145 4.6. Two-period European call option 150 4.7. Multiperiod binomial model 153 4.8. The evolution of the asset prices as a Markov chain 154 4.9.Exercises 158 Chapter 5. Martingales 163 5.1. Introductory notes 163 5.2.Martingales 164 5.3. Optional sampling theorem 169 5.4. Submartingales, supermartingales and martingales convergence theorem 178 5.5.Martingale transforms 182 5.6. Uniform integrability and Doob’s decomposition 184 5.7.The snell envelope 187 5.8.Exercises 190 Chapter 6. Equivalent Martingale Measures, No-Arbitrage and Complete Markets 195 6.1. Introductory notes 195 6.2. Equivalent martingale measure and the Randon-Nikodým derivative process 196 6.3. Finite general markets 204 6.4. Fundamental theorem of asset pricing 215 6.5.Completemarkets andmartingale representation 222 6.6. Finding the equivalent martingale measure 228 6.7.Exercises 238 Chapter 7. American Derivative Securities 241 7.1. Introductory notes 241 7.2.A three-periodAmerican put option 242 7.3. Hedging strategy for an American put option 249 7.4.The algorithm of the American put option 254 7.5.Optimal time for the holder to exercise 255 7.6. American derivatives in general markets 262 7.7. Extending the concept of self-financing strategies 266 7.8.Exercises 269 Chapter 8. Fixed-Income Markets and Interest Rates 273 8.1. Introductory notes 273 8.2. The zero coupon bonds of all maturities 274 8.3. Arbitrage-free family of bond prices 278 8.4. Interest rate process and the term structure of bond prices 282 8.5. The evolution of the interest rate process 290 8.6. Binomial model with normally distributed spread of interest rates 293 8.7. Binomial model with lognormally distributed spread of interest rates 296 8.8. Option arbitrage pricing on zero coupon bonds 298 8.9. Fixed income derivatives 302 8.10. T-period equivalent forward measure 308 8.11. Futures contracts 317 8.12.Exercises 319 Chapter 9. Credit Risk 323 9.1. Introductory notes 323 9.2. Credit ratings and corporate bonds 324 9.3. Credit risk methodologies 326 9.4. Arbitrage pricing of defaultable bonds 327 9.5. Migration process as a Markov chain 330 9.6. Estimation of the real world transition probabilities 334 9.7. Term structure of credit spread and model calibration 337 9.8. Migration process under the real-world probability measure 341 9.9.Exercises 352 Chapter 10. The Heath-Jarrow-Morton Model 355 10.1. Introductory notes 355 10.2. Heath-Jarrow-Morton model 356 10.3. Hedging strategies for zero coupon bonds 362 10.4.Exercises 364 References 365 Appendices 374 Index 395

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Book SynopsisOptimal filtering applied to stationary and non-stationary signals provides the most efficient means of dealing with problems arising from the extraction of noise signals. Moreover, it is a fundamental feature in a range of applications, such as in navigation in aerospace and aeronautics, filter processing in the telecommunications industry, etc. This book provides a comprehensive overview of this area, discussing random and Gaussian vectors, outlining the results necessary for the creation of Wiener and adaptive filters used for stationary signals, as well as examining Kalman filters which are used in relation to non-stationary signals. Exercises with solutions feature in each chapter to demonstrate the practical application of these ideas using MATLAB.Table of ContentsPreface ix Introduction xi Chapter 1. Random Vectors 1 1.1. Definitions and general properties. 1 1.2. Spaces L1 (dP) and L2 (dP) 20 1.3. Mathematical expectation and applications 23 1.4. Second order random variables and vectors. 39 1.5. Linear independence of vectors of L2 (dP) 46 1.6. Conditional expectation (concerning random vectors with density function) 51 1.7. Exercises for Chapter 1 56 Chapter 2. Gaussian Vectors 63 2.1. Some reminders regarding random Gaussian vectors 63 2.2. Definition and characterization of Gaussian vectors 66 2.3. Results relative to independence 68 2.4. Affine transformation of a Gaussian vector 72 2.5. The existence of Gaussian vectors. 74 2.6. Exercises for Chapter 2 84 Chapter 3. Introduction to Discrete Time Processes 93 3.1. Definition 93 3.2. WSS processes and spectral measure 105 3.3. Spectral representation of a WSS process 109 3.4. Introduction to digital filtering 114 3.5. Important example: autoregressive process 127 3.6. Exercises for Chapter 3 132 Chapter 4. Estimation 139 4.1. Position of the problem 139 4.2. Linear estimation 142 4.3. Best estimate – conditional expectation 154 4.4. Example: prediction of an autoregressive process AR (1) 162 4.5. Multivariate processes 163 4.6. Exercises for Chapter 4 172 Chapter 5. The Wiener Filter 177 5.1. Introduction 177 5.2. Resolution and calculation of the FIR filter 179 5.3. Evaluation of the least error 181 5.4. Resolution and calculation of the IIR filter 183 5.5. Evaluation of least mean square error 187 5.6. Exercises for Chapter 5 188 Chapter 6. Adaptive Filtering: Algorithm of the Gradient and the LMS 195 6.1. Introduction 195 6.2. Position of problem 198 6.3. Data representation 200 6.4. Minimization of the cost function 202 6.5. Gradient algorithm 209 6.6. Geometric interpretation 212 6.7. Stability and convergence 216 6.8. Estimation of gradient and LMS algorithm 221 6.9. Example of the application of the LMS algorithm 224 6.10. Exercises for Chapter 6 233 Chapter 7. The Kalman Filter 235 7.1. Position of problem 235 7.2. Approach to estimation 239 7.3. Kalman filtering 243 7.4. Exercises for Chapter 7 261 7.5. Appendices 267 7.6. Examples treated using Matlab software 273 Table of Symbols and Notations 281 Bibliography 283 Index 285

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Book SynopsisQuantitative finance has become these last years a extraordinary field of research and interest as well from an academic point of view as for practical applications. At the same time, pension issue is clearly a major economical and financial topic for the next decades in the context of the well-known longevity risk. Surprisingly few books are devoted to application of modern stochastic calculus to pension analysis. The aim of this book is to fill this gap and to show how recent methods of stochastic finance can be useful for to the risk management of pension funds. Methods of optimal control will be especially developed and applied to fundamental problems such as the optimal asset allocation of the fund or the cost spreading of a pension scheme. In these various problems, financial as well as demographic risks will be addressed and modelled.Table of ContentsPreface xiii Chapter 1. Introduction: Pensions in Perspective 1 1.1. Pension issues 1 1.2. Pension scheme 7 1.3. Pension and risks 11 1.4. The multi-pillar philosophy 14 Chapter 2. Classical Actuarial Theory of Pension Funding 15 2.1. General equilibrium equation of a pension scheme 15 2.2. General principles of funding mechanisms for DB Schemes 21 2.3. Particular funding methods 22 Chapter 3. Deterministic and Stochastic Optimal Control 31 3.1. Introduction 31 3.2. Deterministic optimal control 31 3.3. Necessary conditions for optimality 33 3.4. The maximum principle 42 3.5. Extension to the one-dimensional stochastic optimal control 45 3.6. Examples 52 Chapter 4. Defined Contribution and Defined Benefit Pension Plans 55 4.1. Introduction 55 4.2. The defined benefit method 56 4.3. The defined contribution method 57 4.4. The notional defined contribution (NDC) method 58 4.5. Conclusions 93 Chapter 5. Fair and Market Values and Interest Rate Stochastic Models 95 5.1. Fair value 95 5.2. Market value of financial flows 96 5.3. Yield curve 97 5.4. Yield to maturity for a financial investment and for a bond 99 5.5. Dynamic deterministic continuous time model for an instantaneous interest rate 100 5.6. Stochastic continuous time dynamic model for an instantaneous interest rate 104 5.7. Zero-coupon pricing under the assumption of no arbitrage 114 5.8. Market evaluation of financial flows 130 5.9. Stochastic continuous time dynamic model for asset values 132 5.10. VaR of one asset 136 Chapter 6. Risk Modeling and Solvency for Pension Funds 149 6.1. Introduction 149 6.2. Risks in defined contribution 149 6.3. Solvency modeling for a DC pension scheme 150 6.4. Risks in defined benefit 170 6.5. Solvency modeling for a DB pension scheme 171 Chapter 7. Optimal Control of a Defined Benefit Pension Scheme 181 7.1. Introduction 181 7.2. A first discrete time approach: stochastic amortization strategy 181 7.3. Optimal control of a pension fund in continuous time 194 Chapter 8. Optimal Control of a Defined Contribution Pension Scheme 207 8.1. Introduction 207 8.2. Stochastic optimal control of annuity contracts 208 8.3. Stochastic optimal control of DC schemes with guarantees and under stochastic interest rates 223 Chapter 9. Simulation Models 231 9.1. Introduction231 9.2. The direct method 233 9.3. The Monte Carlo models 250 9.4. Salary lines construction 252 Chapter 10. Discrete Time Semi-Markov Processes (SMP) and Reward SMP 277 10.1. Discrete time semi-Markov processes 277 10.2. DTSMP numerical solutions 280 10.3. Solution of DTHSMP and DTNHSMP in the transient case: a transportation example 284 10.4. Discrete time reward processes 294 10.5. General algorithms for DTSMRWP 304 Chapter 11. Generalized Semi-Markov Non-homogeneous Models for Pension Funds and Manpower Management 307 11.1. Application to pension funds evolution 307 11.2. Generalized non-homogeneous semi-Markov model for manpower management 338 11.3. Algorithms 347 APPENDICES 359 Appendix 1. Basic Probabilistic Tools for Stochastic Modeling 361 Appendix 2. Itô Calculus and Diffusion Processes 397 Bibliography 437 Index 449

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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 SynopsisThis is an introduction to stochastic integration and stochastic differential equations written in an understandable way for a wide audience, from students of mathematics to practitioners in biology, chemistry, physics, and finances. The presentation is based on the naïve stochastic integration, rather than on abstract theories of measure and stochastic processes. The proofs are rather simple for practitioners and, at the same time, rather rigorous for mathematicians. Detailed application examples in natural sciences and finance are presented. Much attention is paid to simulation diffusion processes. The topics covered include Brownian motion; motivation of stochastic models with Brownian motion; Itô and Stratonovich stochastic integrals, Itô’s formula; stochastic differential equations (SDEs); solutions of SDEs as Markov processes; application examples in physical sciences and finance; simulation of solutions of SDEs (strong and weak approximations). Exercises with hints and/or solutions are also provided.Trade Review“Thus, the book is a welcome addition in the effort to make stochastic integration and SDE as accessible as possible to the greater public interested in or in need of using them.” (Mathematical Reviews, 1 February 2013) “If I have a chance to teach (again) a course in stochastic financial modelling, I will definitely choose this to be among two or three sources to use. I have all the reasons to strongly recommend it to anybody in the area of modern stochastic modelling.” (Zentralblatt MATH, 1 December 2012)Table of ContentsPreface 9 Notation 13 Chapter 1. Introduction: Basic Notions of Probability Theory 17 1.1. Probability space 17 1.2. Random variables 21 1.3. Characteristics of a random variable 21 1.4. Types of random variables 23 1.5. Conditional probabilities and distributions 26 1.6. Conditional expectations as random variables 27 1.7. Independent events and random variables 29 1.8. Convergence of random variables 29 1.9. Cauchy criterion 31 1.10. Series of random variables 31 1.11. Lebesgue theorem 32 1.12. Fubini theorem 32 1.13. Random processes 33 1.14. Kolmogorov theorem 34 Chapter 2. Brownian Motion 35 2.1. Definition and properties 35 2.2. White noise and Brownian motion 45 2.3. Exercises 49 Chapter 3. Stochastic Models with Brownian Motion and White Noise 51 3.1. Discrete time 51 3.2. Continuous time 55 Chapter 4. Stochastic Integral with Respect to Brownian Motion 59 4.1. Preliminaries. Stochastic integral with respect to a step process 59 4.2. Definition and properties 69 4.3. Extensions 81 4.4. Exercises 85 Chapter 5. Itô’s Formula 87 5.1. Exercises 94 Chapter 6. Stochastic Differential Equations 97 6.1. Exercises 105 Chapter 7. Itô Processes 107 7.1. Exercises 121 Chapter 8. Stratonovich Integral and Equations 125 8.1. Exercises 136 Chapter 9. Linear Stochastic Differential Equations 137 9.1. Explicit solution of a linear SDE 137 9.2. Expectation and variance of a solution of an LSDE 141 9.3. Other explicitly solvable equations 145 9.4. Stochastic exponential equation 147 9.5. Exercises 153 Chapter 10. Solutions of SDEs as Markov Diffusion Processes 155 10.1. Introduction 155 10.2. Backward and forward Kolmogorov equations 161 10.3. Stationary density of a diffusion process 172 10.4. Exercises 176 Chapter 11. Examples 179 11.1. Additive noise: Langevin equation 180 11.2. Additive noise: general case 180 11.3. Multiplicative noise: general remarks 184 11.4. Multiplicative noise: Verhulst equation 186 11.5. Multiplicative noise: genetic model 189 Chapter 12. Example in Finance: Black–Scholes Model 195 12.1. Introduction: what is an option? 195 12.2. Self-financing strategies 197 12.3. Option pricing problem: the Black–Scholes model 204 12.4. Black–Scholes formula 206 12.5. Risk-neutral probabilities: alternative derivation of Black–Scholes formula 210 12.6. Exercises 214 Chapter 13. Numerical Solution of Stochastic Differential Equations 217 13.1. Memories of approximations of ordinary differential equations 218 13.2. Euler approximation 221 13.3. Higher-order strong approximations 224 13.4. First-order weak approximations 231 13.5. Higher-order weak approximations 238 13.6. Example: Milstein-type approximations 241 13.7. Example: Runge–Kutta approximations 244 13.8. Exercises 249 Chapter 14. Elements of Multidimensional Stochastic Analysis 251 14.1. Multidimensional Brownian motion 251 14.2. Itô’s formula for a multidimensional Brownian motion 252 14.3. Stochastic differential equations 253 14.4. Itô processes 254 14.5. Itô’s formula for multidimensional Itô processes 256 14.6. Linear stochastic differential equations 256 14.7. Diffusion processes 257 14.8. Approximations of stochastic differential equations 259 Solutions, Hints, and Answers 261 Bibliography 271 Index 273

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