Stochastics Books

Book SynopsisThis book explores the martingale approach to the statistical analysis of counting processes, with an emphasis on application of those methods to censored failure time data. Introduced in the 1970s, this approach has proven to be remarkably successful in yielding results about statistical methods for many problems arising in censored data.Trade Review"…a unique source for combining the theory and the application of the survival analysis with censored data." (Technometrics, August 2007)Table of ContentsPreface. 0. The Applied Setting. 1. The Counting Process and Martingale Framework. 2. Local Square Integrable Martingales. 3. Finite Sample Moments and Large Sample Consistency of Tests and Estimators. 4. Censored Data Regression Models and Their Application. 5. Martingale Central Limit Theorem. 6. Large Sample results of the Kaplan-Meier Estimator. 7. Weighted Logrank Statistics. 8. Distribution Theory for Proportional Hazards Regression. Appendix A: Some Results from stieltjes Integration and Probability Theory. Appendix B: An Introduction to Weak convergence. Appendix C: The Martingale Central Limit Theorem: Some Preliminaries. Appendix D: Data. Appendix E: Exercises. Bibliography. Notation. Author Index. Subject Index.

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Book SynopsisA good working knowledge of statistical principles is needed for both the design and analysis of biological experiments and the subsequent handling of the large amounts of data generated if worthwhile, reliable conclusions are to be reached. Practical Statistics for Experimental Biologists, Second Edition provides biologists with a user-friendly, non-technical introduction to the basics of statistics. The book has been thoroughly revised and updated to incorporate: * Worked examples and printouts from MINITAB * Relevant case studies and applications * Further Notes section for background explanations Written by a biologist with extensive experience of applying statistical procedures to experimental systems, this book will be invaluable to undergraduates, postgraduates and researchers in microbiology, immunology, biochemistry, botany, zoology, physiology, pharmacology and pharmacy. Review of the First Edition ...strongly recommended as the current first choicTrade Review"...a refreshing and useful book..." ---- Trends in Plant Science, September 2000Table of ContentsA Simple Experiment in Pipetting. How to Condense the Bulkiness of Data. Are Those Differences Significant? More About Measurement Differences. Awkward-Measurement Data. How to Deal with Count Data. How to Deal with Proportion Data. Correlation and Regression. Dose-Response Lines and Assays. References. Additional Reading. Appendices. Index.

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

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

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

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

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

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

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Book SynopsisBased on a lecture course given at Chalmers University of Technology, this 2002 book is ideal for advanced undergraduate or beginning graduate students. The author first develops the necessary background in probability theory and Markov chains before applying it to study a range of randomized algorithms with important applications in optimization and other problems in computing. Amongst the algorithms covered are the Markov chain Monte Carlo method, simulated annealing, and the recent Propp-Wilson algorithm. This book will appeal not only to mathematicians, but also to students of statistics and computer science. The subject matter is introduced in a clear and concise fashion and the numerous exercises included will help students to deepen their understanding.Trade Review'Has climbing up onto the MCMC juggernaut seemed to require just too much effort? This delightful little monograph provides an effortless way in. The chapters are bite-sized with helpful, do-able exercises (by virtue of strategically placed hints) that complement the text.' Publication of the International Statistical Institute'… a very nice introduction to the modern theory of Markov chain simulation algorithms.' R. E. Maiboroda, Zentralblatt MATH' … extremely elegant. I am sure that students will find great pleasure in using the book - and that teachers will have the same pleasure in using it to prepare a course on the subject.' Mathematics of Computation'This elegant little book is a beautiful introduction to the theory of simulation algorithms, using (discrete) Markov chains (on finite state spaces) … highly recommended to anyone interested in the theory of Markov chain simulation algorithms.' Nieuw Archief voor WiskundeTable of Contents1. Basics of probability theory; 2. Markov chains; 3. Computer simulation of Markov chains; 4. Irreducible and aperiodic Markov chains; 5. Stationary distributions; 6. Reversible Markov chains; 7. Markov chain Monte Carlo; 8. Fast convergence of MCMC algorithms; 9. Approximate counting; 10. Propp-Wilson algorithm; 11. Sandwiching; 12. Propp-Wilson with read once randomness; 13. Simulated annealing; 14. Further reading.

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Book SynopsisThis book is aimed at the trouble with trying to learn about probability. A story of the misconceptions and difficulties civilization overcame in progressing toward probabilistic thinking, Randomness is also a skillful account of what makes the science of probability so daunting in our own day.Trade ReviewClearly, the computation of probabilities is not just an arid game… As Deborah Bennett shows in her excellent little book on the mathematics of chance, the concept has been controversial for thousands of years… [Her] cultured and accessible book goes a long way towards demystifying the science of probability and thereby offers the reader a useful variety of conceptual tools with which to probe the future and illuminate the present. -- Steven Poole * The Guardian *[Randomness] can most easily be described as a brief history of chance… I can cheerfully recommend it to anyone who is a total beginner when it comes to probability, what it means, why it is desperately puzzling, and what it can do for us despite that… It is fascinating to read about the pioneers of probability, such as Pierre Simon de Laplace with his ‘normal distribution’—now more familiar as the notorious bell curve—and Adolphe Quetelet, perhaps the first to realise that there are statistical patterns in human behaviour. And I applaud the blunt reminder that when it comes to the real world the ‘normal’ distribution is actually highly abnormal… My main criticism: it left me wanting more. A sequel, please. -- Ian Stewart * Times Higher Education Supplement *Chances are high that reading this book will clear up your misconceptions about randomness and probabilities. In this very entertaining little book, simply written but intended for careful readers, some of the most common mistakes people make about chance are carefully analyzed. While describing interesting aspects of the mathematics of probability, the author takes frequent detours into the history of humanity’s understanding (and misunderstanding) of the laws of chance, touching on subjects as diverse as chance in decision-making and the fairness of those decisions, gambling and our intuitive understanding of chance, the likelihood of the extremely rare, the existence of true randomness and how computers have helped shape modern thinking about probabilities… An insightful chapter is ‘Chance or Necessity?’ The question is very, very old (determinism versus chaos), and the answer is not clear even today. The author describes the problem beautifully: ‘Is random outcome completely determined, and random only by virtue of our ignorance of the most minute contributing factors?’ Einstein grappled with this conundrum until his death and never ceased to combat the idea that God could conceivably throw dice… Whether well-educated in mathematics or not, people have always been fascinated by randomness and intrigued by the fundamental question of the real nature of randomness, of how you can tell randomness from something that is not. -- J. A. Rial * American Scientist *The great strength of this book is the way it uses history and even prehistory of probability to chart its present territory and cast light on its core point of contention: does true randomness exist in nature, or is it only a psychological artefact?… Bennett’s text…is like a café conversation between likable cognoscenti…nothing could more provoke and excite the reader. -- Simon Ings * New Scientist *In this book, Bennett seeks to account for the centuries-long lapse between early uses of chance in decision making and the more technical studies of probability first undertaken in the seventeenth century. At the same time, she explores the confusions and misunderstandings about probability that persist today. She argues that the notion of randomness played a crucial role in inhibiting conceptual progress in probability and that it also accounts for present-day struggles to come to terms with the subject… Bennett’s book is written in a lucid, engaging style and provides an entertaining introduction to some questions in probability. -- Patti Wilger Hunter * Isis *[A] sharp analysis of the way we assess probability in everyday life. -- Robert Winder * New Statesman & Society *Randomness, by mathematician Deborah J. Bennett, was obviously a labor of love. The result is an interesting book that combines a well-researched, anecdotally presented survey of the history of chance, probability and randomness along with some elementary instruction in probability… It includes a wide-ranging and rich bibliography that reflects the passion of the author for the subject. Anybody interested in gaming, random numbers, the Monte Carlo method and so on will find nice anecdotal descriptions of these topics, together with detailed notes and references to the bibliography for more detailed study. It is a good book to have. -- Stephen Gasiorowicz * Physics Today *In 1996 Charles Hailey and David Helfand reported their calculations of the odds of a commercial airliner being struck by a meteor, in response to speculation about TWA flight 800… They conclude that, in over 30 years of air travel, the probability that a commercial flight would have been hit by a meteor big enough to crash it is 1 in 10. This bit of probability trivia is an indication of human beings continuous struggle to understand probability and chance through the ages, and Deborah Bennett captures the fascination with numbers in this pocket-sized volume. The book is filled with…gems. * Skeptic *This volume is exceptionally readable. It takes away much of the mystery of probability while adding to our sense of wonder. * Wordtrade *The fact that randomness, agency, and holiness can readily displace each other in phenomenological explanations of human action is the central concern that might draw students of consciousness to Bennett’s book. Bennett does an excellent job, explaining and drawing out the major questions that swirl around the randomness–agency–holiness issue. -- T. W. Draper * Journal of Consciousness Studies *[This book] examines randomness and several other notions that were critical to the historical development of probabilistic thinking and that also play an important role in any individual’s understanding of the laws of chance. [It] addresses why, from ancient times to today, people have resorted to chance in making decisions; whether a decision made by random choice is a fair decision; how to figure the odds; what role gambling has played in understanding chance; whether extremely rare events are likely in the long run; why some societies and individuals reject randomness; whether true randomness exists; the view of randomness as uncertainty; why even experts disagree about the many meanings of randomness; and why probability is so counterintuitive. * Journal of Economic Literature *Mathematics is its own language, and sometimes it doesn’t translate readily into other human tongues. But Bennett is brilliantly bilingual, well able to put mathematical concepts into clear, expressive English. Her topic is intrinsically fascinating, for who has not felt buffeted by random events, and who has not sought to see when the wheel of fortune may turn up good luck?… More than an intriguing exploration of a peculiarly fascinating part of mathematics, its coverage, ranging from ancient games of chance to modern probability mind-games, makes it comprehensive as well as compulsively readable. -- Patricia Monaghan * Booklist *A clear and detailed examination of the role of pure chance, with fascinating historical asides. * Kirkus Reviews *A careful and well-written treatment of an intriguing subject. -- Donald Goldsmith, author of The Ultimate EinsteinRandomness tells us about chance by recalling the real history of probability and solving many of its engaging puzzles. Beginners will find themselves welcomed and well led. -- Frederick Mosteller, Harvard UniversityRandomness explains probability and odds in an accessible way. This book puts risk and chance into perspective for the airline passenger and the lottery player alike. -- Henry Petroski, author of Invention by Design: How Engineers Get from Thought to ThingTable of Contents* Chance Encounters * Why Resort to Chance? * When the Gods Played Dice * Figuring the Odds * Thought Games for Gamblers * Chance or Necessity? * Order in Apparent Chaos * Wanted: Random Numbers * Randomness as Uncertainty * Paradoxes in Probability * Notes * Bibliography * Index

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Book SynopsisQuantal Response Equilibrium presents a stochastic theory of games that unites probabilistic choice models developed in psychology and statistics with the Nash equilibrium approach of classical game theory. Nash equilibrium assumes precise and perfect decision making in games, but human behavior is inherently stochastic and people realize that theTrade Review"This book brings together two decades of scholarship on an important model of boundedly rational behavior in strategic decision-making settings. Including numerous important applications in economics, political science, and pure game theory, this unified treatment will be valuable to a wide range of scholars."—Timothy Cason, Purdue University"Quantal response equilibrium is a standard tool for game theorists and has numerous connections to other tools and applications. This book collects and extends existing material on QRE and is a significant contribution to pure, and especially applied, game theory. No other books explicate QRE systematically beyond the introductory level and these authors are the right team for pulling the core material together."—Daniel Friedman, University of California, Santa Cruz"Well-written and easy to follow, this book covers the topic of quantal response equilibrium. The notion of stochastic equilibrium has changed the way game theorists think about long-run and short-run equilibrium. Written by three leading experts, this book is of great importance to researchers in economic theory and political science, and to graduate students."—David K. Levine, European University InstituteTable of Contents*Frontmatter, pg. i*Contents, pg. v*Preface, pg. ix*1. Introduction and Background, pg. 1*2. Quantal Response Equilibrium in Normal-Form Games, pg. 10*3. Quantal Response Equilibrium in Extensive-Form Games, pg. 63*4. Heterogeneity, pg. 88*5. Dynamics and Learning, pg. 112*6. QRE as a Structural Model for Estimation, pg. 141*7. Applications to Game Theory, pg. 161*8. Applications to Political Science, pg. 206*9. Applications to Economics, pg. 248*10. Epilogue: Some Thoughts about Future Research, pg. 281*References, pg. 291*Index, pg. 301

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Book SynopsisExplains diffusive motion and its relation to both nonrelativistic quantum theory and quantum field theory. This book shows how diffusive motion concepts lead to a radical reexamination of the structure of mathematical analysis.Table of Contents*FrontMatter, pg. i*Contents, pg. vii*Preface, pg. ix*Chapter One. Introduction: Diffusive Motion and Where It Leads, pg. 1*Chapter Two. Hypercontractivity, Logarithmic Sobolev Inequalities, and Applications: A Survey of Surveys, pg. 45*Chapter Three. Ed Nelson's Work in Quantum Theory, pg. 75*Chapter Four Symanzik, Nelson, and Self-Avoiding Walk, pg. 95*Chapter Five. Stochastic Mechanics: A Look Back and a Look Ahead, pg. 117*Chapter Six. Current Trends in Optimal Transportation: A Tribute to Ed Nelson, pg. 141*Chapter Seven. Internal Set Theory and Infinitesimal Random Walks, pg. 157*Chapter Eight. Nelson's Work on Logic and Foundations and Other Reflections on the Foundations of Mathematics, pg. 183*Chapter Nine. Some Musical Groups: Selected Applications of Group Theory in Music, pg. 209*Chapter Ten. Afterword, pg. 229*Appendix A. Publications by Edward Nelson, pg. 233*Index, pg. 241

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Book SynopsisThis book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the Martingale problem aTable of ContentsPreface xi 1 Introduction 1*1.1 Generalized Kimura Diffusions 3 *1.2 Model Problems 5 *1.3 Perturbation Theory 9 *1.4 Main Results 10 *1.5 Applications in Probability Theory 13 *1.6 Alternate Approaches 14 *1.7 Outline of Text 16 *1.8 Notational Conventions 20 I Wright-Fisher Geometry and the Maximum Principle 23 2 Wright-Fisher Geometry 25*2.1 Polyhedra and Manifolds with Corners 25 *2.2 Normal Forms and Wright-Fisher Geometry 29 3 Maximum Principles and Uniqueness Theorems 34*3.1 Model Problems 34 *3.2 Kimura Diffusion Operators on Manifolds with Corners 35 *3.3 Maximum Principles for theHeat Equation 45 II Analysis of Model Problems 49 4 The Model Solution Operators 51*4.1 The Model Problemin 1-dimension 51 *4.2 The Model Problem in Higher Dimensions 54 *4.3 Holomorphic Extension 59 *4.4 First Steps Toward Perturbation Theory 62 5 Degenerate Holder Spaces 64*5.1 Standard Holder Spaces 65 *5.2 WF-Holder Spaces in 1-dimension 66 6 Holder Estimates for the 1-dimensional Model Problems 78*6.1 Kernel Estimates for Degenerate Model Problems 80 *6.2 Holder Estimates for the 1-dimensional Model Problems 89 *6.3 Propertiesof the Resolvent Operator 103 7 Holder Estimates for Higher Dimensional CornerModels 107*7.1 The Cauchy Problem 109 *7.2 The Inhomogeneous Case 122 *7.3 The Resolvent Operator 135 8 Holder Estimates for Euclidean Models 137*8.1 Holder Estimates for Solutions in the Euclidean Case 137 *8.2 1-dimensional Kernel Estimates 139 9 Holder Estimates for General Models 143*9.1 The Cauchy Problem 145 *9.2 The Inhomogeneous Problem 149 *9.3 Off-diagonal and Long-time Behavior 166 *9.4 The Resolvent Operator 169 III Analysis of Generalized Kimura Diffusions 179 10 Existence of Solutions 181*10.1 WF-Holder Spaces on a Manifold with Corners 182 *10.2 Overview of the Proof 187 *10.3 The Induction Argument 191 *10.4 The Boundary Parametrix Construction 194 *10.5 Solution of the Homogeneous Problem 205 *10.6 Proof of the Doubling Theorem 208 *10.7 The Resolvent Operator and C0-Semi-group 209 *10.8 Higher Order Regularity 211 11 The Resolvent Operator 218*11.1 Construction of the Resolvent 220 *11.2 Holomorphic Semi-groups 229 *11.3 DiffusionsWhere All Coefficients Have the Same Leading Homogeneity 230 12 The Semi-group on C0(P) 235*12.1 The Domain of the Adjoint 237 *12.2 The Null-space of L 240 *12.3 Long Time Asymptotics 243 *12.4 Irregular Solutions of the Inhomogeneous Equation 247 A Proofs of Estimates for the Degenerate 1-d Model 251* A.1 Basic Kernel Estimates 252 * A.2 First Derivative Estimates 272 * A.3 Second Derivative Estimates 278 * A.4 Off-diagonal and Large-t Behavior 291 Bibliography 301 Index 305

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Book SynopsisPreface * 1. Preliminaries: Discrete Index Sets and/or Discrete State Spaces * 2. Markov Chains * 3. Renewal Theory * 4. Point Processes * 5. Continuous Time Markov Chains * 6. Brownian Motion * 7. The General Random Walk * References * IndexTrade Review"Definitely the best textbook for a second course in probability now available. Written with excruciating lucidity, and with an excellent choice of exercises." —Gian-Carlo Rota, The Bulletin of Mathematics Books "In summary, Resnick has succeeded [in writing] a very fine textbook which will become popular among students as well as among professors preparing an introductory course on stochastic processes." —Internationale Mathematische Nachrichten "A splendid book to bring home the value and importance of stochastic processes. Highly recommended." —Choice "There are so many good introductory texts on [stochastic processes] that one can hardly hope to write a better or more attractive one. This book, however, convinced the reviewer that it very likely that the Adventures will beocme a widely used, popular first year graduate text on stochastic processes. The book is flexible, the motivations of deep theories are clear, the examples and exercises are interesting." ---Zentralblatt MATHTable of ContentsPreface * 1. Preliminaries: Discrete Index Sets and/or Discrete State Spaces * 2. Markov Chains * 3. Renewal Theory * 4. Point Processes * 5. Continuous Time Markov Chains * 6. Brownian Motion * 7. The General Random Walk * References * Index

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Book Synopsis1 Measure Theory.- 2 Integration.- 3 Fourier Analysis.- Appendix A Metric Spaces.- Appendix C A Non-Measurable Subset of the Interval (0, 1].- References.Trade Review"…the text is user friendly to the topics it considers and should be very accessible…Instructors and students of statistical measure theoretic courses will appreciate the numerous informative exercises; helpful hints or solution outlines are given with many of the problems. All in all, the text should make a useful reference for professionals and students."—The Journal of the American Statistical AssociationTable of Contents1 Measure Theory.- 2 Integration.- 3 Fourier Analysis.- Appendix A Metric Spaces.- Appendix C A Non-Measurable Subset of the Interval (0, 1].- References.

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Book SynopsisInstead, A Probability Path is designed for those requiring a deep understanding of advanced probability for their research in statistics, applied probability, biology, operations research, mathematical finance and engineering.Trade ReviewFrom the reviews:“This introduction to measure-theoretic probability is intended for students whose primary interest is not mathematics but statistics, engineering, biology, or finance. The book is a welcome reprint in paperback … . The book’s pace … is ‘quick and disciplined’.” (William J. Satzer, MAA Reviews, March, 2014)Table of Contents1 Sets and Events.- 2 Probability Spaces.- 3 Random Variables, Elements and Measurable Maps.- 4 Independence.- 5 Integration and Expectation.- 6 Convergence Concepts.- 7 Laws of Large Numbers and Sums of Independent Random Variables.- 8 Convergence in Distribution.- 9 Characteristic Functions and the Central Limit Theorem.- 10 Martingales.- Index.- References.

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Book SynopsisProbability theory has become a convenient language and a useful tool in many areas of modern analysis. This book intends to explore part of this connection concerning the relations between Brownian motion on a manifold and analytical aspects of differential geometry. It begins with a review of stochastic differential equations on Euclidean space.Table of ContentsIntroduction Stochastic differential equations and diffusions Basic stochastic differential geometry Brownian motion on manifolds Brownian motion and heat kernel Short-time asymptotics Further applications Brownian motion and analytic index theorems Analysis on path spaces Notes and comments General notations Bibliography Index.

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Book SynopsisBased on lectures given at the CBMS Workshop on the Combinatorics of Large Sparse Graphs, this work presents fresh perspectives in graph theory and helps to contribute to a sound scientific foundation for our understanding of discrete networks that permeate the information age.Table of ContentsGraph theory in the information age Old and new concentration inequalities A generative model--the preferential attachment scheme Duplication models for biological networks Random graphs with given expected degrees The rise of the giant component Average distance and the diameter Eigenvalues of the adjacency matrix of $G(\mathbf{w})$ The semi-circle law for $G(\mathbf{w})$ Coupling on-line and off-line analyses of random graphs The configuration model for power law graphs The small world phenomenon in hybrid graphs Bibliography Index.

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Book SynopsisPresents Brownian motion and deals with stochastic integrals and differentials, including Ito lemma. This book is devoted to topics of stochastic integral equations and stochastic integral equations on smooth manifolds. It is suitable for graduate students and researchers interested in probability, stochastic processes, and their applications.Table of ContentsBrownian motion Stochastic integrals and differentials Stochastic integral equations $(d=1)$ Stochastic integral equations $(d\geq2)$ References Subject index Errata.

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Book SynopsisA collection of survey and research papers that gives a glance of the profound consequences of Molchanov's contributions in stochastic differential equations, spectral theory for deterministic and random operators, localization and intermittency, mathematical physics and optics, and other topics.Table of ContentsTransition asymptotics for reaction-diffusion in random media by G. B. Arous, S. Molchanov, and A. Ramirez Extreme value theory for random exponentials by L. V. Bogachev Singular continuous and dense point spectrum for sparse trees with finite dimensions by J. Breuer Some new estimates on the spectral shift function associated with random Schrodinger operators by J.-M. Combes, P. D. Hislop, and F. Klopp On phase transitions and limit theorems for homopolymers by M. Cranston and S. Molchanov Asymptotics of the Poincare functions by G. Derfel, P. J. Grabner, and F. Vogl Hamiltonian extension and eigenfunctions for a time dispersive dissipative string by A. Figotin and J. Schenker Localization at low energies for attractive Poisson random Schrodinger operators by F. Germinet, P. D. Hislop, and A. Klein On the influence of random perturbations on the propagation of waves described by a periodic Schrodinger operator by Y. A. Godin, S. Molchanov, and B. Vainberg Spectral theory of 1-D Schrodinger operators with unbounded potentials by A. Gordon, J. L. Holt, and S. Molchanov Fermi-Dirac generators and tests for randomness by A. Gordon, S. Molchanov, and J. Quinn The spectral problem, substitutions and iterated monodromy by R. Grigorchuk, D. Savchuk, and Z. Sunic On scattering of solitons for wave equation coupled to a particle by V. Imaykin, A. Komech, and B. Vainberg Purely absolutely continuous spectrum for some random Jacobi matrices by U. Kaluzhny and Y. Last The parabolic Anderson model and its universality classes by W. Konig An inverse problem for Gibbs fields by L. Koralov Hierarchical Anderson model by E. Kritchevski Integral representations of solutions of periodic elliptic equations by P. Kuchment Inverse spectral problems for Schrodinger operators with energy depending potentials by A. Laptev, R. Shterenberg, and V. Sukhanov Theory of point processes and some basic notions in energy level statistics by N. Minami On the law of addition of random matrices: Covariance and the central limit theorem for traces of resolvent by L. Pastur and V. Vasilchuk Green's functions of generalized Lapalcians by P. Poulin Orthogonal polynomials with exponentially decaying recursion coefficients by B. Simon Poisson statistics for eigenvalues: From random Schrodinger operators to random CMV matrices by M. Stoiciu.

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

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Book SynopsisA systematic, self-contained treatment of the theory of stochastic differential equations in infinite dimensional spaces. Included is a discussion of Schwartz spaces of distributions in relation to probability theory and infinite dimensional stochastic analysis, as well as the random variables and stochastic processes that take values in infinite dimensional spaces.

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Book SynopsisThis book is intended for students and practitioners of robotics working with noisy sensor data to estimate state variables. New edition highlights include a new chapter on variational inference and new sections on adaptive covariance estimation and on inertial navigation as well as a primer on matrix calculus.Trade Review'This book provides a timely, concise, and well-scoped introduction to state estimation for robotics. It complements existing textbooks by giving a balanced presentation of estimation theoretic and geometric tools and discusses how these tools can be used to solve common estimation problems arising in robotics. It also strikes an excellent balance between theory and motivating examples.' Luca Carlone, IEEE Control Systems MagazineTable of ContentsAcronyms and abbreviations; Notation; Foreword to first edition; Foreword to second edition; 1. Introduction; Part I. Estimation Machinery: 2. Primer on probability theory; 3. Linear-Gaussian estimation; 4. Nonlinear non-Gaussian estimation; 5. Handling nonidealities in estimation; 6. Variational inference; Part II. Three-Dimensional Machinery: 7. Primer on three-dimensional geometry; 8. Matrix lie groups; Part III. Applications: 9. Pose estimation problems; 10. Pose-and-point estimation problems; 11. Continuous-time estimation; Appendix A: matrix primer; Appendix B: rotation and pose extras; Appendix C: miscellaneous extras; Appendix D: solutions to exercises; References; Index.

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Book SynopsisAn Advanced Course in Probability and Stochastic Processes provides a modern and rigorous treatment of probability theory and stochastic processes at an upper undergraduate and graduate level. Starting with the foundations of measure theory, this book introduces the key concepts of probability theory in an accessible way, providing full proofs and extensive examples and illustrations. Fundamental stochastic processes such as Gaussian processes, Poisson random measures, Lévy processes, Markov processes, and Itô processes are presented and explored in considerable depth, showcasing their many interconnections. Special attention is paid to martingales and the Wiener process and their central role in the treatment of stochastic integrals and stochastic calculus. This book includes many exercises, designed to test and challenge the reader and expand their skillset. An Advanced Course in Probability and Stochastic Processes is meant for students and researchers who have a soTable of Contents1. Measure Theory 2. Probability 3. Convergence 4. Conditioning 5. Martingales 6. Wiener and Brownian Motion Processes 7. Itô Calculus Appendix A. Selected Solutions Appendix B. Function Spaces Appendix C. Existence of the Lebesgue Measure Index

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Book SynopsisSince the publication of the first edition of this book, the area of mathematical finance has grown rapidly, with financial analysts using more sophisticated mathematical concepts, such as stochastic integration, to describe the behavior of markets and to derive computing methods. Maintaining the lucid style of its popular predecessor, Introduction to Stochastic Calculus Applied to Finance, Second Edition incorporates some of these new techniques and concepts to provide an accessible, up-to-date initiation to the field. New to the Second EditionComplements on discrete models, including Rogers'' approach to the fundamental theorem of asset pricing and super-replication in incomplete markets Discussions on local volatility, Dupire''s formula, the change of numéraire techniques, forward measures, and the forward Libor model A new chapter on credit risk modeling An extension of the chapter on simulTrade ReviewThe second edition of this book provides a concise and accessible introduction to the probabilistic techniques needed to understand the most widely used financial models. This edition incorporates many new techniques and concepts to be used to describe the behavior of financial markets. … the solutions obtained using SciLab for computer experiments are available at http://cermics.enpc.fr/~bl/scilab/ These experiments were well designed by the authors based on their teaching and research experience and were found to be effective in communicating these concepts and ideas and enhancing the understanding of readers. … a solid introduction to stochastic approaches used in the financial world. The authors cover many key finance topics … . The book can be used as a reference text by researchers and graduate students in financial mathematics. It also is ideal reading material for practicing financial analysts and consultants using mathematical models for finance.—Technometrics, May 2009, Vol. 51, No. 2 Table of ContentsDiscrete-Time Models. Optimal Stopping Problem and American Options. Brownian Motion and Stochastic Differential Equations. The Black-Scholes Model. Option Pricing and Partial Differential Equations. Interest Rate Models. Asset Models with Jumps. Credit Risk Models. Simulation and Algorithms for Financial Models. Appendix. Bibliography. Index.

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Book SynopsisThis third edition contains new problems and exercises, new proofs, expanded material on financial mathematics, financial engineering, and mathematical statistics, and a final chapter on the history of probability theory.Trade Review“I think this would be an excellent text by itself for an advanced course in probability.” (Allen Stenger, MAA Reviews, October 26, 2019)Table of ContentsPreface.- Chapter 4: Sequences and Sums of Independent Random Variables.- Chapter 5: Stationary (Strict Sense) Random Sequences and Ergodic Theory.- Chapter 6: Stationary (Wide Sense) Random Sequences: L2-Theory.- Chapter 7: Martingales.- Chapter 8: Markov Chains.- Historical of Bibliographical Notes (Chapters 4-8).- References.- Index.- Index of Symbols.

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Book SynopsisCovering point process theory, random geometric graphs and coverage processes, this rigorous introduction to stochastic geometry enables the effective analysis of wireless network performance across all possible network configurations, promoting good design choices for future wireless architectures and protocols that reduce interference effects.Trade Review'This book is a welcome addition to the rapidly developing area of applications of stochastic geometric models to telecommunications.' Ilya S. Molchanov, American Mathematical SocietyTable of ContentsPart I. Point Process Theory: 1. Introduction; 2. Description of point processes; 3. Point process models; 4. Sums and products over point processes; 5. Interference and outage in wireless networks; 6. Moment measures of point processes; 7. Marked point processes; 8. Conditioning and Palm theory; Part II. Percolation, Connectivity and Coverage: 9. Introduction; 10. Bond and site percolation; 11. Random geometric graphs and continuum percolation; 12. Connectivity; 13. Coverage; Appendix: introduction to R.

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Book SynopsisThis book is dedicated to the mathematical study of two-dimensional statistical hydrodynamics and turbulence, described by the 2D NavierâStokes system with a random force. The authors' main goal is to justify the statistical properties of a fluid's velocity field u(t,x) that physicists assume in their work. They rigorously prove that u(t,x) converges, as time grows, to a statistical equilibrium, independent of initial data. They use this to study ergodic properties of u(t,x) â proving, in particular, that observables f(u(t,.)) satisfy the strong law of large numbers and central limit theorem. They also discuss the inviscid limit when viscosity goes to zero, normalising the force so that the energy of solutions stays constant, while their Reynolds numbers grow to infinity. They show that then the statistical equilibria converge to invariant measures of the 2D Euler equation and study these measures. The methods apply to other nonlinear PDEs perturbed by random forces.Table of Contents1. Preliminaries; 2. Two-dimensional Navier–Stokes equations; 3. Uniqueness of stationary measure and mixing; 4. Ergodicity and limiting theorems; 5. Inviscid limit; 6. Miscellanies; 7. Appendix; 8. Solutions to some exercises.

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Book SynopsisStochastic systems provide powerful abstract models for a variety of important real-life applications: for example, power supply, traffic flow, data transmission. They (and the real systems they model) are often subject to phase transitions, behaving in one way when a parameter is below a certain critical value, then switching behaviour as soon as that critical value is reached. In a real system, we do not necessarily have control over all the parameter values, so it is important to know how to find critical points and to understand system behaviour near these points. This book is a modern presentation of the 'semimartingale' or 'Lyapunov function' method applied to near-critical stochastic systems, exemplified by non-homogeneous random walks. Applications treat near-critical stochastic systems and range across modern probability theory from stochastic billiards models to interacting particle systems. Spatially non-homogeneous random walks are explored in depth, as they provide prototyTrade Review'This is another impressive volume in the prestigious `Cambridge Tracts in Mathematics' series … The authors of this book are well-known for their long standing and well-recognized contributions to this area of research. Besides their own results published over the last two decades, the authors cover all significant achievements up to date … It is remarkable to see detailed `Bibliographical notes' at the end of each chapter. The authors have done a great job by providing valuable information about the historical development of any topic treated in this book. We find extremely interesting facts, stories and references. All this makes the book more than interesting to read and use.' Jordan M. Stoyanov, Zentralblatt MATH'This book gives a comprehensive account of the study of random walks with spatially non-homogeneous transition kernels. The main theme is to study recurrence versus transience and moments of passage times, as well as path asymptotics, by constructing suitable Lyapunov functions, which define semi-martingales when composed with the random walk. Of special interest are the Lamperti processes, which are stochastic processes on [0, ∞) with drift vanishing asymptotically on the order of 1/x as the space variable x tends to infinity. … Each chapter ends with detailed bibliographical notes.' Rongfeng Sun, Mathematical ReviewsTable of Contents1. Introduction; 2. Semimartingale approach and Markov chains; 3. Lamperti's problem; 4. Many-dimensional random walks; 5. Heavy tails; 6. Further applications; 7. Markov chains in continuous time; Glossary of named assumptions; Bibliography; Index.

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Book SynopsisNow in its second edition, this book gives a systematic and self-contained presentation of basic results on stochastic evolution equations in infinite dimensional, typically Hilbert and Banach, spaces. Thoroughly updated, it also includes two brand new chapters surveying recent developments in the area.Trade ReviewReview of the first edition: 'The exposition is excellent and readable throughout, and should help bring the theory to a wider audience.' Daniel L. Ocone, Stochastics and Stochastic ReportsReview of the first edition: '… a welcome contribution to the rather new area of infinite dimensional stochastic evolution equations, which is far from being complete, so it should provide both a useful background and motivation for further research.' Yuri Kifer, The Annals of ProbabilityReview of the first edition: '… an excellent book which covers a large part of stochastic evolution equations with clear proofs and a very interesting analysis of their properties … In my opinion this book will become an indispensable tool for everybody working on stochastic evolution equations and related areas.' P. Kotelenez, American Mathematical SocietyTable of ContentsPreface; Introduction; Part I. Foundations: 1. Random variables; 2. Probability measures; 3. Stochastic processes; 4. Stochastic integral; Part II. Existence and Uniqueness: 5. Linear equations with additive noise; 6. Linear equations with multiplicative noise; 7. Existence and uniqueness for nonlinear equations; 8. Martingale solutions; 9. Markov property and Kolmogorov equation; 10. Absolute continuity and Girsanov theorem; 11. Large time behavior of solutions; 12. Small noise asymptotic; 13. Survey of specific equations; 14. Some recent developments; Appendix A. Linear deterministic equations; Appendix B. Some results on control theory; Appendix C. Nuclear and Hilbert–Schmidt operators; Appendix D. Dissipative mappings; Bibliography; Index.

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Book SynopsisNow in its second edition, this popular textbook on game theory is unrivalled in the breadth of its coverage, the thoroughness of technical explanations and the number of worked examples included. Covering non-cooperative and cooperative games, this introduction to game theory includes advanced chapters on auctions, games with incomplete information, games with vector payoffs, stable matchings and the bargaining set. This edition contains new material on stochastic games, rationalizability, and the continuity of the set of equilibrium points with respect to the data of the game. The material is presented clearly and every concept is illustrated with concrete examples from a range of disciplines. With numerous exercises, and the addition of a solution manual for instructors with this edition, the book is an extensive guide to game theory for undergraduate through graduate courses in economics, mathematics, computer science, engineering and life sciences, and will also serve as useful reTrade ReviewPraise for first edition: 'This is the book for which the world has been waiting for decades: a definitive, comprehensive account of the mathematical theory of games, by three of the world's biggest experts on the subject. Rigorous yet eminently readable, deep yet comprehensible, replete with a large variety of important real-world applications, it will remain the standard reference in game theory for a very long time.' Robert Aumann, Nobel Laureate in Economics, The Hebrew University of JerusalemPraise for first edition: 'Without any sacrifice on the depth or the clarity of the exposition, this book is amazing in its breadth of coverage of the important ideas of game theory. It covers classical game theory, including utility theory, equilibrium refinements and belief hierarchies; classical cooperative game theory, including the core, Shapley value, bargaining set and nucleolus; major applications, including social choice, auctions, matching and mechanism design; and the relevant mathematics of linear programming and fixed point theory. The comprehensive coverage combined with the depth and clarity of exposition makes it an ideal book not only to learn game theory from, but also to have on the shelves of working game theorists.' Ehud Kalai, Kellogg School of Management, Northwestern UniversityPraise for first edition: 'The best and the most comprehensive textbook for advanced courses in game theory.' David Schmeidler, Ohio State University and Tel Aviv UniversityPraise for first edition: 'There are quite a few good textbooks on game theory now, but for rigor and breadth this one stands out.' Eric S. Maskin, Nobel Laureate in Economics, Harvard University, MassachusettsPraise for first edition: 'This textbook provides an exceptionally clear and comprehensive introduction to both cooperative and noncooperative game theory. It deftly combines a rigorous exposition of the key mathematical results with a wealth of illuminating examples drawn from a wide range of subjects. It is a tour de force.' Peyton Young, University of OxfordPraise for first edition: 'This is a wonderful introduction to game theory, written in a way that allows it to serve both as a text for a course and as a reference … The book is written by leading figures in the field [whose] broad view of the field suffuses the material.' Joe Halpern, Cornell University, New YorkTable of Contents1. The game of chess; 2. Utility theory; 3. Extensive-form games; 4. Strategic-form games; 5. Mixed strategies; 6. Behavior strategies and Kuhn's theorem; 7. Equilibrium refinements; 8. Correlated equilibria; 9. Games with incomplete information and common priors; 10. Games with incomplete information: the general model; 11. The universal belief space; 12. Auctions; 13. Repeated games; 14. Repeated games with vector payoffs; 15. Social choice; 16. Bargaining games; 17. Coalitional games with transferable utility; 18. The core; 19. The Shapley value; 20. The bargaining set; 21. The nucleolus; 22. Stable matching; 23. Appendices.

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Book SynopsisOne of the first books to provide in-depth and systematic application of finite element methods to the field, Stochastic Structural Dynamics presents and illustrates direct integration methods for analyzing the statistics of the response of structures to stochastic loads.Table of ContentsDedication xi Preface xiii Acknowledgements xv 1. Introduction 1 1.1 Displacement Formulation Based Finite Element Method 2 1.2 Element Equations of Motion for Temporally and Spatially Stochastic Systems 13 1.3 Hybrid Stress Based Element Equations of Motion 14 1.4 Incremental Variational Principle and Mixed Formulation Based Nonlinear Element Matrices 18 1.5 Constitutive Relations and Updating of Configurations and Stresses 36 1.6 Concluding Remarks 48 References 49 2. Spectral Analysis and Response Statistics of Linear Structural Systems 53 2.1 Spectral Analysis 53 2.2 Evolutionary Spectral Analysis 56 2.3 Evolutionary Spectra of Engineering Structures 60 2.4 Modal Analysis and Time-Dependent Response Statistics 76 2.5 Response Statistics of Engineering Structures 79 References 94 3. Direct Integration Methods for Linear Structural Systems 97 3.1 Stochastic Central Difference Method 97 3.2 Stochastic Central Difference Method with Time Co-ordinate Transformation 100 3.3 Applications 102 3.4 Extended Stochastic Central Difference Method and Narrow-band Force Vector 114 3.5 Stochastic Newmark Family of Algorithms 122 References 128 4. Modal Analysis and Response Statistics of Quasi-linear Structural Systems 131 4.1 Modal Analysis of Temporally Stochastic Quasi-linear Systems 131 4.2 Response Analysis Based on Melosh-Zienkiewicz-Cheung Bending Plate Finite Element 141 4.3 Response Analysis Based on High Precision Triangular Plate Finite Element 156 4.4 Concluding Remarks 166 References 166 5. Direct Integration Methods for Response Statistics of Quasi-linear Structural Systems 169 5.1 Stochastic Central Difference Method for Quasi-linear Structural Systems 169 5.2 Recursive Covariance Matrix of Displacements of Cantilever Pipe Containing Turbulent Fluid 174 5.3 Quasi-linear Systems under Narrow-band Random Excitations 184 5.4 Concluding Remarks 188 References 190 6. Direct Integration Methods for Temporally Stochastic Nonlinear Structural Systems 191 6.1 Statistical Linearization Techniques 191 6.2 Symplectic Algorithms of Newmark Family of Integration Schemes 194 6.3 Stochastic Central Difference Method with Time Co-ordinate Transformation and Adaptive Time Schemes 199 6.4 Outline of steps in computer program 211 6.5 Large Deformations of Plate and Shell Structures 213 6.6 Concluding Remarks 224 References 226 7. Direct Integration Methods for Temporally and Spatially Stochastic Nonlinear Structural Systems 231 7.1 Perturbation Approximation Techniques and Stochastic Finite Element Methods 232 7.2 Stochastic Central Difference Methods for Temporally and Spatially Stochastic Nonlinear Systems 241 7.3 Finite Deformations of Spherical Shells with Large Spatially Stochastic Parameters 251 7.4 Closing Remarks 255 References 257 Appendices 1A Mass and Stiffness Matrices of Higher Order Tapered Beam Element 261 1B Consistent Stiffness Matrix of Lower Order Triangular Shell Element 267 1B.1 Inverse of Element Generalized Stiffness Matrix 267 1B.2 Element Leverage Matrices 268 1B.3 Element Component Stiffness Matrix Associated with Torsion 271 References 276 1C Consistent Mass Matrix of Lower Order Triangular Shell Element 277 Reference 280 2A Eigenvalue Solution 281 References 282 2B Derivation of Evolutionary Spectral Densities and Variances of Displacements 283 2B.1 Evolutionary Spectral Densities Due to Exponentially Decaying Random Excitations 283 2B.2 Evolutionary Spectral Densities Due to Uniformly Modulated Random Excitations 286 2B.3 Variances of Displacements 288 References 297 2C Time-dependent Covariances of Displacements 299 2D Covariances of Displacements and Velocities 311 2E Time-dependent Covariances of Velocities 317 2F Cylindrical Shell Element Matrices 323 3A Deterministic Newmark Family of Algorithms 327 Reference 331 Index 333

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Book Synopsis Unifies existing and emerging concepts concerning multi-objective control and stochastic control with engineering-oriented phenomena Establishes a unified theoretical framework for control and filtering problems for a class of discrete-time nonlinear stochastic systems with consideration to performance Includes case studies of several nonlinear stochastic systems Investigates the phenomena of incomplete information, including missing/degraded measurements, actuator failures and sensor saturations Considers both time-invariant systems and time-varying systems Exploits newly developed techniques to handle the emerging mathematical and computational challenges Table of ContentsPreface vii Acknowledgements ix List of Abbreviations xi 1 Introduction 1 1.1 Analysis and Synthesis of Nonlinear Stochastic Systems 2 1.1.1 Nonlinear Systems 3 1.1.2 Stochastic Systems 4 1.2 Multi-Objective Control and Filtering with Variance Constraints 5 1.2.1 Covariance Control Theory 5 1.2.2 Multiple Performance Requirements 7 1.2.3 Design Techniques for Nonlinear Stochastic Systems with Variance Constraints 9 1.2.4 A Special Case of Multi-Objective Design: Mixed H2/H1 Control/Filtering 11 1.3 Outline 12 2 Robust H1 Control with Variance Constraints 17 2.1 Problem Formulation 18 2.2 Stability, H1 Performance and Variance Analysis 20 2.2.1 Stability, H1 Performance Analysis 21 2.2.2 Variance Analysis 23 2.3 Robust Controller Design 27 2.4 Numerical Example 30 2.5 Summary 33 3 Robust Mixed H2=H1 Filtering 41 3.1 System Description and Problem Formulation 42 3.2 Algebraic Characterizations for Robust H2=H1 Filtering 44 3.2.1 Robust H2 Filtering 44 3.2.2 Robust H1 Filtering 50 3.3 Robust H2=H1 Filter Design Techniques 51 3.4 An Illustrative Example 60 3.5 Summary 62 4 Filtering with Missing Measurements 63 4.1 Problem Formulation 64 4.2 Stability and Variance Analysis 67 4.3 Robust Filter Design 71 4.4 Numerical Example 75 4.5 Summary 78 5 Robust Fault-Tolerant Control 87 5.1 Problem Formulation 88 5.2 Stability and Variance Analysis 90 5.3 Robust Controller Design 92 5.4 Numerical Example 98 5.5 Summary 103 6 Robust H2 SMC 105 6.1 The System Model 106 6.2 Robust H2 Sliding Mode Control 107 6.2.1 Switching Surface 107 6.2.2 Performances of the Sliding Motion 108 6.2.3 Computational Algorithm 114 6.3 Sliding Mode Controller 115 6.4 Numerical Example 116 6.5 Summary 118 7 Dissipative Control with Degraded Measurements 125 7.1 Problem Formulation 126 7.2 Stability, Dissipativity and Variance Analysis 129 7.3 Observer-Based Controller Design 134 7.3.1 Solvability of Multi-Objective Control Problem 134 7.3.2 Computational Algorithm 139 7.4 Numerical Example 140 7.5 Summary 142 8 Variance-Constrained H1 Control with Multiplicative Noises 145 8.1 Problem Formulation 146 8.2 Stability, H1 Performance, Variance Analysis 147 8.2.1 Stability 148 8.2.2 H1 performance 150 8.2.3 Variance analysis 152 8.3 Robust State Feedback Controller Design 153 8.4 A Numerical Example 156 8.5 Summary 157 9 Robust Finite-Horizon H1 Control 159 9.1 Problem Formulation 160 9.2 Performance Analysis 162 9.2.1 H1 Performance 162 9.2.2 Variance Analysis 164 9.3 Robust Finite Horizon Controller Design 167 9.4 Numerical Example 171 9.5 Summary 173 10 Finite-Horizon Filtering with Degraded Measurements 177 10.1 Problem Formulation 178 10.2 Performance Analysis 181 10.2.1 H1 Performance Analysis 181 10.2.2 System Covariance Analysis 186 10.3 Robust Filter Design 187 10.4 Numerical Example 190 10.5 Summary 191 11 Mixed H2=H1 Control with Randomly Occurring Nonlinearities: the Finite-Horizon Case 197 11.1 Problem Formulation 199 11.2 H1 Performance 200 11.3 Mixed H2=H1 Controller Design 204 11.3.1 State-Feedback Controller Design 204 11.3.2 Computational Algorithm 207 11.4 Numerical Example 207 11.5 Summary 211 12 Finite-Horizon H2=H1 Control of MJSs with Sensor Failures 213 12.1 Problem Formulation 214 12.2 H1 Performance 216 12.3 Mixed H2=H1 Controller Design 220 12.3.1 Controller Design 220 12.3.2 Computational Algorithm 224 12.4 Numerical Example 224 12.5 Summary 227 13 Finite-Horizon Control with ROSF 229 13.1 Problem Formulation 230 13.2 H1 And Covariance Performances Analysis 234 13.2.1 H1 Performance 234 13.2.2 Covariance Analysis 238 13.3 Robust Finite-Horizon Controller Design 240 13.3.1 Controller Design 240 13.3.2 Computational Algorithm 243 13.4 Numerical Example 243 13.5 Summary 244 14 Finite-Horizon H2=H1 Control with Actuator Failures 247 14.1 Problem Formulation 248 14.2 H1 Performance 251 14.3 Multi-Objective Controller Design 253 14.3.1 Controller Design 253 14.3.2 Computational Algorithm 256 14.4 Numerical Example 257 14.5 Summary 259 15 Conclusions and Future Topics 261 References 263

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Book SynopsisA beginner's guide to stochastic growth modeling The chief advantage of stochastic growth models over deterministic models is that they combine both deterministic and stochastic elements of dynamic behaviors, such as weather, natural disasters, market fluctuations, and epidemics. This makes stochastic modeling a powerful tool in the hands of practitioners in fields for which population growth is a critical determinant of outcomes. However, the background requirements for studying SDEs can be daunting for those who lack the rigorous course of study received by math majors. Designed to be accessible to readers who have had only a few courses in calculus and statistics, this book offers a comprehensive review of the mathematical essentials needed to understand and apply stochastic growth models. In addition, the book describes deterministic and stochastic applications of population growth models including logistic, generalized logistic, Gompertz, negative exponentiTrade Review"An indispensable resource for students and practitioners with limited exposure tomathematics and statistics, Stochastic Differential Equations: An Introduction withApplications in Population Dynamics Modeling is an excellent fit for advanced under-graduates and beginning graduate students, as well as practitioners who need a gentleintroduction to SDEs" Mathematical Reviews, October 2017Table of ContentsDedication x Preface xi Symbols and Abbreviations xiii 1 Mathematical Foundations 1: Point-Set Concepts, Set and Measure Functions, Normed Linear Spaces, and Integration 1 1.1 Set Notation and Operations 1 1.1.1 Sets and Set Inclusion 1 1.1.2 Set Algebra 2 1.2 Single-Valued Functions 4 1.3 Real and Extended Real Numbers 6 1.4 Metric Spaces 7 1.5 Limits of Sequences 8 1.6 Point-Set Theory 10 1.7 Continuous Functions 12 1.8 Operations on Sequences of Sets 13 1.9 Classes of Subsets of Ω 15 1.9.1 Topological Space 15 1.9.2 σ-Algebra of Sets and the Borel σ-Algebra 15 1.10 Set and Measure Functions 17 1.10.1 Set Functions 17 1.10.2 Measure Functions 18 1.10.3 Outer Measure Functions 19 1.10.4 Complete Measure Functions 21 1.10.5 Lebesgue Measure 21 1.10.6 Measurable Functions 23 1.10.7 Lebesgue Measurable Functions 26 1.11 Normed Linear Spaces 27 1.11.1 Space of Bounded Real-Valued Functions 27 1.11.2 Space of Bounded Continuous Real-Valued Functions 28 1.11.3 Some Classical Banach Spaces 29 1.12 Integration 31 1.12.1 Integral of a Non-negative Simple Function 32 1.12.2 Integral of a Non-negative Measurable Function Using Simple Functions 33 1.12.3 Integral of a Measurable Function 33 1.12.4 Integral of a Measurable Function on a Measurable Set 34 1.12.5 Convergence of Sequences of Functions 35 2 Mathematical Foundations 2: Probability, Random Variables, and Convergence of Random Variables 37 2.1 Probability Spaces 37 2.2 Probability Distributions 42 2.3 The Expectation of a Random Variable 49 2.3.1 Theoretical Underpinnings 49 2.3.2 Computational Considerations 50 2.4 Moments of a Random Variable 52 2.5 Multiple Random Variables 54 2.5.1 The Discrete Case 54 2.5.2 The Continuous Case 59 2.5.3 Expectations and Moments 63 2.5.4 The Multivariate Discrete and Continuous Cases 69 2.6 Convergence of Sequences of Random Variables 72 2.6.1 Almost Sure Convergence 73 2.6.2 Convergence in Lp,p>0 73 2.6.3 Convergence in Probability 75 2.6.4 Convergence in Distribution 75 2.6.5 Convergence of Expectations 76 2.6.6 Convergence of Sequences of Events 78 2.6.7 Applications of Convergence of Random Variables 79 2.7 A Couple of Important Inequalities 80 Appendix 2.A The Conditional Expectation E(X|Y) 81 3 Mathematical Foundations 3: Stochastic Processes, Martingales, and Brownian Motion 85 3.1 Stochastic Processes 85 3.1.1 Finite-Dimensional Distributions of a Stochastic Process 86 3.1.2 Selected Characteristics of Stochastic Processes 88 3.1.3 Filtrations of A 89 3.2 Martingales 91 3.2.1 Discrete-Time Martingales 91 3.2.1.1 Discrete-Time Martingale Convergence 93 3.2.2 Continuous-Time Martingales 96 3.2.2.1 Continuous-Time Martingale Convergence 97 3.2.3 Martingale Inequalities 97 3.3 Path Regularity of Stochastic Processes 98 3.4 Symmetric Random Walk 99 3.5 Brownian Motion 100 3.5.1 Standard Brownian Motion 100 3.5.2 BM as a Markov Process 104 3.5.3 Constructing BM 106 3.5.3.1 BM Constructed from N(0, 1) Random Variables 106 3.5.3.2 BM as the Limit of Symmetric Random Walks 108 3.5.4 White Noise Process 109 Appendix 3.A Kolmogorov Existence Theorem: Another Look 109 Appendix 3.B Nondifferentiability of BM 110 4 Mathematical Foundations 4: Stochastic Integrals, Itô’s Integral, Itô’s Formula, and Martingale Representation 113 4.1 Introduction 113 4.2 Stochastic Integration: The Itô Integral 114 4.3 One-Dimensional Itô Formula 120 4.4 Martingale Representation Theorem 126 4.5 Multidimensional Itô Formula 127 Appendix 4.A Itô’s Formula 129 Appendix 4.B Multidimensional Itô Formula 130 5 Stochastic Differential Equations 133 5.1 Introduction 133 5.2 Existence and Uniqueness of Solutions 134 5.3 Linear SDEs 136 5.3.1 Strong Solutions to Linear SDEs 137 5.3.2 Properties of Solutions 147 5.3.3 Solutions to SDEs as Markov Processes 152 5.4 SDEs and Stability 154 Appendix 5.A Solutions of Linear SDEs in Product Form (Evans, 2013; Gard, 1988) 159 5.A.1 Linear Homogeneous Variety 159 5.A.2 Linear Variety 161 Appendix 5.B Integrating Factors and Variation of Parameters 162 5.B.1 Integrating Factors 163 5.B.2 Variation of Parameters 164 6 Stochastic Population Growth Models 167 6.1 Introduction 167 6.2 A Deterministic Population Growth Model 168 6.3 A Stochastic Population Growth Model 169 6.4 Deterministic and Stochastic Logistic Growth Models 170 6.5 Deterministic and Stochastic Generalized Logistic Growth Models 174 6.6 Deterministic and Stochastic Gompertz Growth Models 177 6.7 Deterministic and Stochastic Negative Exponential Growth Models 179 6.8 Deterministic and Stochastic Linear Growth Models 181 6.9 Stochastic Square-Root Growth Model with Mean Reversion 182 Appendix 6.A Deterministic and Stochastic Logistic Growth Models with an Allee Effect 184 Appendix 6.B Reducible SDEs 189 7 Approximation and Estimation of Solutions to Stochastic Differential Equations 193 7.1 Introduction 193 7.2 Iterative Schemes for Approximating SDEs 194 7.2.1 The EM Approximation 194 7.2.2 Strong and Weak Convergence of the EM Scheme 196 7.2.3 The Milstein (Second-Order) Approximation 196 7.3 The Lamperti Transformation 199 7.4 Variations on the EM and Milstein Schemes 203 7.5 Local Linearization Techniques 205 7.5.1 The Ozaki Method 205 7.5.2 The Shoji–Ozaki Method 207 7.5.3 The Rate of Convergence of the Local Linearization Method 211 Appendix 7.A Stochastic Taylor Expansions 212 Appendix 7.B The EM and Milstein Discretizations 217 7.B.1 The EM Scheme 217 7.B.2 The Milstein Scheme 218 Appendix 7.C The Lamperti Transformation 219 8 Estimation of Parameters of Stochastic Differential Equations 221 8.1 Introduction 221 8.2 The Transition Probability Density Function Is Known 222 8.3 The Transition Probability Density Function Is Unknown 227 8.3.1 Parameter Estimation via Approximation Methods 228 8.3.1.1 The EM Routine 228 8.3.1.2 The Ozaki Routine 230 8.3.1.3 The SO Routine 233 Appendix 8.A The ML Technique 235 Appendix 8.B The Log-Normal Probability Distribution 238 Appendix 8.C The Markov Property, Transitional Densities, and the Likelihood Function of the Sample 239 Appendix 8.D Change of Variable 241 Appendix A: A Review of Some Fundamental Calculus Concepts 245 Appendix B: The Lebesgue Integral 259 Appendix C: Lebesgue–Stieltjes Integral 261 Appendix D: A Brief Review of Ordinary Differential Equations 263 References 275 Index 279

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