Stochastics Books

Book SynopsisThe fourth edition of this widely used textbook on pricing and hedging of financial derivatives now also includes dynamic equilibrium theory and continues to combine sound mathematical principles with economic applications. Concentrating on the probabilistic theory of continuous time arbitrage pricing of financial derivatives, including stochastic optimal control theory and optimal stopping theory, Arbitrage Theory in Continuous Time is designed for graduate students in economics and mathematics, and combines the necessary mathematical background with a solid economic focus. It includes a solved example for every new technique presented, contains numerous exercises, and suggests further reading in each chapter. All concepts and ideas are discussed, not only from a mathematics point of view, but with lots of intuitive economic arguments.In the substantially extended fourth edition Tomas Björk has added completely new chapters on incomplete markets, treating such topics as the Esscher trTrade ReviewReview from previous edition This book is one of the best of a large number of new books on mathematical and probabilistic models in finance, positioned between the books by Hull and Duffie on a mathematical scale...This is a highly reasonable book and strikes a balance between mathematical development and intuitive explanation. * Short Book Reviews *Table of Contents1: Introduction I. Discrete Time Models 2: The Binomial Model 3: A More General One period Model II. Stochastic Calculus 4: Stochastic Integrals 5: Stochastic Differential Equations III. Arbitrage Theory 6: Portfolio Dynamics 7: Arbitrage Pricing 8: Completeness and Hedging 9: A Primer on Incomplete Markets 10: Parity Relations and Delta Hedging 11: The Martingale Approach to Arbitrage Theory 12: The Mathematics of the Martingale Approach 13: Black-Scholes from a Martingale Point of View 14: Multidimensional Models: Martingale Approach 15: Change of Numeraire 16: Dividends 17: Forward and Futures Contracts 18: Currency Derivatives 19: Bonds and Interest Rates 20: Short Rate Models 21: Martingale Models for the Short Rate 22: Forward Rate Models 23: LIBOR Market Models 24: Potentials and Positive Interest IV. Optimal Control and Investment Theory 25: Stochastic Optimal Control 26: Optimal Consumption and Investment 27: The Martingale Approach to Optimal Investment 28: Optimal Stopping Theory and American Options V. Incomplete Markets 29: Incomplete Markets 30: The Esscher Transform and the Minimal Martingale Measure 31: Minimizing f-divergence 32: Portfolio Optimization in Incomplete Markets 33: Utility Indifference Pricing and Other Topics 34: Good Deal Bounds VI. Dynamic Equilibrium Theory 35: Equilibrium Theory: A Simple Production Model 36: The Cox-Ingersoll-Ross Factor Model 37: The Cox-Ingersoll-Ross Interest Rate Model 38: Endowment Equilibrium: Unit Net Supply

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This book sheds new light on stochastic calculus, the branch of mathematics that is most widely applied in financial engineering and mathematical finance. The first book to introduce pathwise formulae for the stochastic integral, it provides a simple but rigorous treatment of the subject, including a range of advanced topics. The book discusses in-depth topics such as quadratic variation, Ito formula, and Emery topology. The authors briefly addresses continuous semi-martingales to obtain growth estimates and study solution of a stochastic differential equation (SDE) by using the technique of random time change. Later, by using Metivier–Pellaumail inequality, the solutions to SDEs driven by general semi-martingales are discussed. The connection of the theory with mathematical finance is briefly discussed and the book has extensive treatment on the representation of martingales as stochastic integrals and a second fundamental theorem of asset pricing. Intended for undergraduate- and beginning graduate-level students in the engineering and mathematics disciplines, the book is also an excellent reference resource for applied mathematicians and statisticians looking for a review of the topic.

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Book SynopsisSuitable for self study Use real examples and real data sets that will be familiar to the audience Introduction to the bootstrap is included – this is a modern method missing in many other books Trade ReviewFrom the reviews: "[the material is] superbly motivated with interest-grabbing examples... exercises excellent and plentiful." Edward Williams, University of Michigan-Dearborn, USA "... it is a notoriously hard task to introduce probability and statistics with a mix of intuition and mathematics to keep students motivated. Therefore, I very much welcome this book and recommend it as course material." Sara van de Geer, Leiden University, The Netherlands "This textbook provides a well-written first course in probability and statistics...It is a book that has been written based on the long teaching experience of the authors and I would certainly recommend it for university coursework." Short Book Reviews of the International Statistical Institute, December 2005 "This book has numerous quick exercises to give direct feedback to the students. … A website at www.springeronline.com/978-1-85233-896-1 gives access to the data files used in the text … . This will be a key text for undergraduates in computer science, physics, mathematics, chemistry, biology and business studies who are studying a mathematical statistics course, and also for more intensive engineering statistics courses for undergraduates in all engineering subjects." (Rainer Beedgen, Zentralblatt MATH, Vol. 1079, 2006) "The book is designed for a one-semester introductory course in probability and statistics basics for engineering students. … It can also be used by students in other more mathematically oriented majors such as applied mathematics with more emphasis on the mathematics and additional coverage in topics such as combinatorics, conditional expectation, and generating functions. … More elaborate exercises and real datasets are given at the end of each chapter." (Arthur B. Yeh, Technometrics, Vol. 49 (3), August, 2007)Table of ContentsWhy probability and statistics?.- Outcomes, events, and probability.- Conditional probability and independence.- Discrete random variables.- Continuous random variables.- Simulation.- Expectation and variance.- Computations with random variables.- Joint distributions and independence.- Covariance and correlation.- More computations with more random variables.- The Poisson process.- The law of large numbers.- The central limit theorem.- Exploratory data analysis: graphical summaries.- Exploratory data analysis: numerical summaries.- Basic statistical models.- The bootstrap.- Unbiased estimators.- Efficiency and mean squared error.- Maximum likelihood.- The method of least squares.- Confidence intervals for the mean.- More on confidence intervals.- Testing hypotheses: essentials.- Testing hypotheses: elaboration.- The t-test.- Comparing two samples.

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Book SynopsisThis second edition textbook offers a practical introduction to probability for undergraduates at all levels with different backgrounds and views towards applications. Calculus is a prerequisite for understanding the basic concepts, however the book is written with a sensitivity to students’ common difficulties with calculus that does not obscure the thorough treatment of the probability content. The first six chapters of this text neatly and concisely cover the material traditionally required by most undergraduate programs for a first course in probability. The comprehensive text includes a multitude of new examples and exercises, and careful revisions throughout. Particular attention is given to the expansion of the last three chapters of the book with the addition of one entirely new chapter (9) on ’Finding and Comparing Estimators.’ The classroom-tested material presented in this second edition forms the basis for a second course introducing mathematical statistics.Table of ContentsProbability Space.- Conditional probabilities.- Discrete random variables.- Binomial random variables.- Poisson random variables.- Simulations of discrete random variables.- Combinatorics.- Continuous random variables.- The sample average and sample.- Estimating and testing proportions.- Estimating and testing means.- Small samples.- Chi-squared tests.- Design of experiments.- The cumulative distribution function.- Continuous joint distributions.- Covariance and independence.- Conditional distribution and expectation.- The bivariate normal distribution.- Sums of Bernoulli random variables.- Coupling random variables.- The moment generating function.- The chi-squared, Student and F distributions.- Sampling from a normal distribution.- Finding estimators.- Comparing estimators.- Best unbiased estimators.- Bayes’ estimator.- Multiple linear regression.- List of common discrete distributions.- List of common continuous distributions.- Further reading.- Normal table.- Student table.- Chi-squared table.- Index.

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Book SynopsisTable of Contents1. Introduction to Probability Theory 2. Random Variables 3. Conditional Probability and Conditional Expectation 4. Markov Chains 5. The Exponential Distribution and the Poisson Process 6. Continuous-Time Markov Chains 7. Renewal Theory and Its Applications 8. Queueing Theory 9. Reliability Theory 10. Brownian Motion and Stationary Processes 11. Simulation 12. Coupling 13. Martingales

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Book SynopsisTable of ContentsCHAPTER 1 Introduction to statistics CHAPTER 2 Descriptive statistics CHAPTER 3 Elements of probability CHAPTER 4 Random variables and expectation CHAPTER 5 Special random variables CHAPTER 6 Distributions of sampling statistics CHAPTER 7 Parameter estimation CHAPTER 8 Hypothesis testing CHAPTER 9 Regression CHAPTER 10 Analysis of variance CHAPTER 11 Goodness of fit tests and categorical data analysis CHAPTER 12 Nonparametric hypothesis tests CHAPTER 13 Quality control CHAPTER 14 Life testing CHAPTER 15 Simulation, bootstrap statistical methods, and permutation tests CHAPTER 16 Machine learning and big data

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Book SynopsisThis book presents the theory of rational decisions involving the selection of stopping times in observed discrete-time stochastic processes, both by single and multiple decision-makers. Readers will become acquainted with the models, strategies, and applications of these models.It begins with an examination of selected models framed as stochastic optimization challenges, emphasizing the critical role of optimal stopping times in sequential statistical procedures. The authors go on to explore models featuring multiple stopping and shares on leading applications, particularly focusing on change point detection, selection problems, and the nuances of behavioral ecology. In the following chapters, an array of perspectives on model strategies is presented, elucidating their interpretation and the methodologies underpinning their genesis. Essential notations and definitions are introduced, examining general theorems about solution existence and structure, with an intricate analysi

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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 xi Introduction xiii Chapter 1. Random Vectors 1 1.1. Definitions and general properties 1 1.2. Spaces L1(dP) and L2(dP) 20 1.2.1. Definitions 20 1.2.2. Properties 22 1.3. Mathematical expectation and applications 23 1.3.1. Definitions 23 1.3.2. Characteristic functions of a random vector 34 1.4. Second order random variables and vectors 39 1.5. Linear independence of vectors of L2(dP) 47 1.6. Conditional expectation (concerning random vectors with density function) 51 1.7. Exercises for Chapter 1 57 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 85 Chapter 3. Introduction to Discrete Time Processes 93 3.1. Definition 93 3.2. WSS processes and spectral measure 105 3.2.1. Spectral density 106 3.3. Spectral representation of a WSS process 110 3.3.1. Problem 110 3.3.2. Results 111 3.3.2.1. Process with orthogonal increments and associated measurements 111 3.3.2.2. Wiener stochastic integral 113 3.3.2.3. Spectral representation 114 3.4. Introduction to digital filtering 115 3.5. Important example: autoregressive process 128 3.6. Exercises for Chapter 3 134 Chapter 4. Estimation 141 4.1. Position of the problem 141 4.2. Linear estimation 144 4.3. Best estimate – conditional expectation 156 4.4. Example: prediction of an autoregressive process AR (1) 165 4.5. Multivariate processes 166 4.6. Exercises for Chapter 4 175 Chapter 5. The Wiener Filter 181 5.1. Introduction 181 5.1.1. Problem position 182 5.2. Resolution and calculation of the FIR filter 183 5.3. Evaluation of the least error 185 5.4. Resolution and calculation of the IIR filter 186 5.5. Evaluation of least mean square error 190 5.6. Exercises for Chapter 5 191 Chapter 6. Adaptive Filtering: Algorithm of the Gradient and the LMS 197 6.1. Introduction 197 6.2. Position of problem 199 6.3. Data representation 202 6.4. Minimization of the cost function 204 6.4.1. Calculation of the cost function 208 6.5. Gradient algorithm 211 6.6. Geometric interpretation 214 6.7. Stability and convergence 218 6.8. Estimation of gradient and LMS algorithm 222 6.8.1. Convergence of the algorithm of the LMS 225 6.9. Example of the application of the LMS algorithm 225 6.10. Exercises for Chapter 6 234 Chapter 7. The Kalman Filter 237 7.1. Position of problem 237 7.2. Approach to estimation 241 7.2.1. Scalar case 241 7.2.2. Multivariate case 244 7.3. Kalman filtering 245 7.3.1. State equation 245 7.3.2. Observation equation 246 7.3.3. Innovation process 248 7.3.4. Covariance matrix of the innovation process 248 7.3.5. Estimation 250 7.3.6. Riccati’s equation 258 7.3.7. Algorithm and summary 260 7.4. Exercises for Chapter 7 262 Table of Symbols and Notations 281 Bibliography 283 Index 285

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Book SynopsisThis graduate-level textbook covers modelling, programming and analysis of stochastic computer simulation experiments, including the mathematical and statistical foundations of simulation and why it works. The book is rigorous and complete, but concise and accessible, providing all necessary background material. Object-oriented programming of simulations is illustrated in Python, while the majority of the book is programming language independent. In addition to covering the foundations of simulation and simulation programming for applications, the text prepares readers to use simulation in their research. A solutions manual for end-of-chapter exercises is available for instructors.Table of ContentsChapter 1: Why Do We Simulate.- Chapter 2: Simulation Programming: Quick Start.- Chapter 3: Examples.- Chapter 4: Simulation Programming with PythonSim.- Chapter 5: Three Views of Simulation.- Chapter 6: Simulation Input.- Chapter 7: Simulation Output.- Chapter 8: Experiment Design and Analysis.- Chapter 9: Simulation Optimization and Sensitivity.- Chapter 10: Simulation for Research.- References.- Index.

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Book SynopsisThis book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Itô’s formula, the optional stopping theorem and Girsanov’s theorem, are treated in detail alongside many illustrative examples. The book also contains an introduction to Markov processes, with applications to solutions of stochastic differential equations and to connections between Brownian motion and partial differential equations. The theory of local times of semimartingales is discussed in the last chapter.Since its invention by Itô, stochastic calculus has proven to be one of the most important techniques of modern probability theory, and has been used in the most recent theoretical advances as well as in applications to other fields such as mathematical finance. Brownian Motion, Martingales, and Stochastic Calculus provides a strong theoretical background to the reader interested in such developments.Beginning graduate or advanced undergraduate students will benefit from this detailed approach to an essential area of probability theory. The emphasis is on concise and efficient presentation, without any concession to mathematical rigor. The material has been taught by the author for several years in graduate courses at two of the most prestigious French universities. The fact that proofs are given with full details makes the book particularly suitable for self-study. The numerous exercises help the reader to get acquainted with the tools of stochastic calculus.Trade Review“‘The aim of this book is to provide a rigorous introduction to the theory of stochastic calculus for continuous semi-martingales putting a special emphasis on Brownian motion.’ … If the reader has the background and needs a rigorous treatment of the subject this book would be a good choice. Le Gall writes clearly and gets to the point quickly … .” (Richard Durrett, MAA Reviews, March, 2017) “The purpose of this book is to provide concise but rigorous introduction to the theory of stochastic calculus for continuous semimartingales, putting a special emphasis on Brownian motion. … The book is written very clearly, it is interesting both for its construction and maintenance, mostly it is self-contained. It can be recommended to everybody who wants to study stochastic calculus, including those who is interested to its applications in other fields.” (Yuliya S. Mishura, zbMATH, 2017)Table of ContentsGaussian variables and Gaussian processes.- Brownian motion.- Filtrations and martingales.- Continuous semimartingales.- Stochastic integration.- General theory of Markov processes.- Brownian motion and partial differential equations.- Stochastic differential equations.- Local times.- The monotone class lemma.- Discrete martingales.- References.

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Book SynopsisProbability is an area of mathematics of tremendous contemporary importance across all aspects of human endeavour. This book is a compact account of the basic features of probability and random processes at the level of first and second year mathematics undergraduates and Masters'' students in cognate fields. It is suitable for a first course in probability, plus a follow-up course in random processes including Markov chains.A special feature is the authors'' attention to rigorous mathematics: not everything is rigorous, but the need for rigour is explained at difficult junctures. The text is enriched by simple exercises, together with problems (with very brief hints) many of which are taken from final examinations at Cambridge and Oxford. The first eight chapters form a course in basic probability, being an account of events, random variables, and distributions - discrete and continuous random variables are treated separately - together with simple versions of the law of large numbersTable of ContentsPART A BASIC PROBABILITY; PART B FURTHER PROBABILITY

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Book SynopsisTaken literally, the title All of Statistics is an exaggeration. But in spirit, the title is apt, as the book does cover a much broader range of topics than a typical introductory book on mathematical statistics. This book is for people who want to learn probability and statistics quickly. It is suitable for graduate or advanced undergraduate students in computer science, mathematics, statistics, and related disciplines. The book includes modern topics like nonparametric curve estimation, bootstrapping, and clas sification, topics that are usually relegated to follow-up courses. The reader is presumed to know calculus and a little linear algebra. No previous knowledge of probability and statistics is required. Statistics, data mining, and machine learning are all concerned with collecting and analyzing data. For some time, statistics research was con ducted in statistics departmeTrade ReviewWinner of the 2005 DeGroot Prize.From the reviews:"Presuming no previous background in statistics and described by the author as "demanding" yet "understandable because the material is as intuitive as possible" (p. viii), this certainly would be my choice of textbook if I was required to learn mathematical statistics again for a couple of semesters." Technometrics, August 2004"This book should be seriously considered as a text for a theoretical statsitics course for non-majors, and perhaps even for majors...The coverage of emerging and important topics is timely and welcomed...you should have this book on your desk as a reference to nothing less than 'All of Statistics.'" Biometrics, December 2004"Although All of Statistics is an ambitious title, this book is a concise guide, as the subtitle suggests....I recommend it to anyone who has an interest in learning something new about statistical inference. There is something here for everyone." The American Statistician, May 2005"As the title of the book suggests, ‘All of Statistics’ covers a wide range of statistical topics. … The number of topics covered in this book is vast … . The greatest strength of this book is as a first point of reference for a wide range of statistical methods. … I would recommend this book as a useful and interesting introduction to a large number of statistical topics for non-statisticians and also as a useful reference book for practicing statisticians." (Matthew J. Langdon, Journal of Applied Statistics, Vol. 32 (1), January, 2005)"This book was written specifically to give students a quick but sound understanding of modern statistics, and its coverage is very wide. … The book is extremely well done … ." (N. R. Draper, Short Book Reviews, Vol. 24 (2), 2004)"This is most definitely a book about mathematical statistics. It is full of theorems and proofs … . Presuming no previous background in statistics … this certainly would be my choice of textbook if I was required to learn mathematical statistics again for a couple of semesters." (Eric R. Ziegel, Technometrics, Vol. 46 (3), August, 2004)"The author points out that this book is for those who wish to learn probability and statistics quickly … . this book will serve as a guideline for instructors as to what should constitute a basic education in modern statistics. It introduces many modern topics … . Adequate references are provided at the end of each chapter which the instructor will be able to use profitably … ." (Arup Bose, Sankhya, Vol. 66 (3), 2004)"The amount of material that is covered in this book is impressive. … the explanations are generally clear and the wide range of techniques that are discussed makes it possible to include a diverse set of examples … . The worked examples are complemented with numerous theoretical and practical exercises … . is a very useful overview of many areas of modern statistics and as such will be very useful to readers who require such a survey. Library copies would also see plenty of use." (Stuart Barber, Journal of the Royal Statistical Society, Series A – Statistics in Society, Vol. 168 (1), 2005)Table of ContentsProbability.- Random Variables.- Expectation.- Inequalities.- Convergence of Random Variables.- Models, Statistical Inference and Learning.- Estimating the CDF and Statistical Functionals.- The Bootstrap.- Parametric Inference.- Hypothesis Testing and p-values.- Bayesian Inference.- Statistical Decision Theory.- Linear and Logistic Regression.- Multivariate Models.- Inference about Independence.- Causal Inference.- Directed Graphs and Conditional Independence.- Undirected Graphs.- Loglinear Models.- Nonparametric Curve Estimation.- Smoothing Using Orthogonal Functions.- Classification.- Probability Redux: Stochastic Processes.- Simulation Methods.

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Book SynopsisThe fourth edition of this successful text provides an introduction to probability and random processes, with many practical applications. It is aimed at mathematics undergraduates and postgraduates, and has four main aims.US BL To provide a thorough but straightforward account of basic probability theory, giving the reader a natural feel for the subject unburdened by oppressive technicalities. BE BL To discuss important random processes in depth with many examples.BE BL To cover a range of topics that are significant and interesting but less routine. BE BL To impart to the beginner some flavour of advanced work.BE UE OP The book begins with the basic ideas common to most undergraduate courses in mathematics, statistics, and science. It ends with material usually found at graduate level, for example, Markov processes, (including Markov chain Monte Carlo), martingales, queues, diffusions, (including stochastic calculus with Itô''s formula), renewals, stationary processes (including the ergodic theorem), and option pricing in mathematical finance using the Black-Scholes formula. Further, in this new revised fourth edition, there are sections on coupling from the past, Lévy processes, self-similarity and stability, time changes, and the holding-time/jump-chain construction of continuous-time Markov chains. Finally, the number of exercises and problems has been increased by around 300 to a total of about 1300, and many of the existing exercises have been refreshed by additional parts. The solutions to these exercises and problems can be found in the companion volume, One Thousand Exercises in Probability, third edition, (OUP 2020).CPTrade ReviewFeatures of PRP include brief but helpful motivational introductions to each subsection, and copious references to historical applications. To aid navigation, definitions, theorems and other key results are highlighted, using three different colours. The tone throughout is rigorous but the touch is human ... * Owen Toller, The Mathematical Gazette *Since its first appearance in 1982 Probability and Random Processes has been a landmark book on the subject and has become mandatory reading for any mathematician wishing to understand chance. It is aimed mainly at final-year honours students and graduate students, but it goes beyond this level, and all serious mathematicians and academic libraries should own a copy ... the companion book of exercises is cleverly conceived and ... forms a perfect complement to the main text. * Times Higher Education Supplement *Review from previous edition...a full and comprehensive account of (almost all) the probability theory and stochastic processes one could hope to teach to undergraduates.... As well as its masterful coverage of the material, the book has many appealing stylistic features ... extremely valuable in finding good proofs of theorems which are dealt with rather cursorily in other textbooks. * The Mathematical Gazette *One of the strong features of the book is its large collection of interesting exercises, which has been greatly expanded in this new edition so that there are now over one thousand. These are conveniently collected together in a separate volume that includes full solutions. * Biometrics *Table of Contents1: Events and their probabilities 2: Random variables and their distributions 3: Discrete random variables 4: Continuous random variables 5: Generating functions and their applications 6: Markov chains 7: Convergence of random variables 8: Random processes 9: Stationary processes 10: Renewals 11: Queues 12: Martingales 13: Diffusion processes

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Book SynopsisStochastic processes are tools used widely by statisticians and researchers working in the mathematics of finance. This book for self-study provides a detailed treatment of conditional expectation and probability, a topic that in principle belongs to probability theory, but is essential as a tool for stochastic processes. The book centers on exercises as the main means of explanation.Trade ReviewThis book fulfils its aim of providing good and interesting material for advanced undergraduate study. The Times Higher Education Supplement This is probably one of the best books to begin learning about the sometimes complex topic of stochastic calculus and stochastic processes from a more mathematical approach. Some literature are often accused of unnecessarily complicating the subject when applied to areas of finance. With this book you are allowed to explore the rigorous side of stochastic calculus, yet maintain a physical insight of what is going on. The authors have concentrated on the most important and useful topics that are encountered in common physical and financial systems www.quantnotes.com Table of Contents1. Review of Probability.- 1.1 Events and Probability.- 1.2 Random Variables.- 1.3 Conditional Probability and Independence.- 1.4 Solutions.- 2. Conditional Expectation.- 2.1 Conditioning on an Event.- 2.2 Conditioning on a Discrete Random Variable.- 2.3 Conditioning on an Arbitrary Random Variable.- 2.4 Conditioning on a ?-Field.- 2.5 General Properties.- 2.6 Various Exercises on Conditional Expectation.- 2.7 Solutions.- 3. Martingales in Discrete.- 3.1 Sequences of Random Variables.- 3.2 Filtrations.- 3.3 Martingales.- 3.4 Games of Chance.- 3.5 Stopping Times.- 3.6 Optional Stopping Theorem.- 3.7 Solutions.- 4. Martingale Inequalities and Convergence.- 4.1 Doob’s Martingale Inequalities.- 4.2 Doob’s Martingale Convergence Theorem.- 4.3 Uniform Integrability and L1 Convergence of Martingales.- 4.4 Solutions.- 5. Markov Chains.- 5.1 First Examples and Definitions.- 5.2 Classification of States.- 5.3 Long-Time Behaviour of Markov Chains: General Case.- 5.4 Long-Time Behaviour of Markov Chains with Finite State Space.- 5.5 Solutions.- 6. Stochastic Processes in Continuous Time.- 6.1 General Notions.- 6.2 Poisson Process.- 6.2.1 Exponential Distribution and Lack of Memory.- 6.2.2 Construction of the Poisson Process.- 6.2.3 Poisson Process Starts from Scratch at Time t.- 6.2.4 Various Exercises on the Poisson Process.- 6.3 Brownian Motion.- 6.3.1 Definition and Basic Properties.- 6.3.2 Increments of Brownian Motion.- 6.3.3 Sample Paths.- 6.3.4 Doob’s Maximal L2 Inequality for Brownian Motion.- 6.3.5 Various Exercises on Brownian Motion.- 6.4 Solutions.- 7. Itô Stochastic Calculus.- 7.1 Itô Stochastic Integral: Definition.- 7.2 Examples.- 7.3 Properties of the Stochastic Integral.- 7.4 Stochastic Differential and Itô Formula.- 7.5 Stochastic Differential Equations.- 7.6 Solutions.

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Book SynopsisThe book provides an introduction to advanced topics in stochastic processes and related stochastic analysis, and combines them with a sound presentation of the fundamentals of financial mathematics. It is wide-ranging in content, while at the same time placing much emphasis on good readability, motivation, and explanation of the issues covered. Financial mathematical topics are first introduced in the context of discrete time processes and then transferred to continuous-time models. The basic construction of the stochastic integral and the associated martingale theory provide fundamental methods of the theory of stochastic processes for the construction of suitable stochastic models of financial mathematics, e.g. using stochastic differential equations. Central results of stochastic analysis such as the Itô formula, Girsanov's theorem and martingale representation theorems are of fundamental importance in financial mathematics, e.g. for the risk-neutral valuation formula (Black-Scholes formula) or the question of the hedgeability of options and the completeness of market models. Chapters on the valuation of options in complete and incomplete markets and on the determination of optimal hedging strategies conclude the range of topics.Advanced knowledge of probability theory is assumed, in particular of discrete-time processes (martingales, Markov chains) and continuous-time processes (Brownian motion, Lévy processes, processes with independent increments, Markov processes). The book is thus suitable for advanced students as a companion reading and for instructors as a basis for their own courses.This book is a translation of the original German 1st edition Stochastische Prozesse und Finanzmathematik by Ludger Rüschendorf, published by Springer-Verlag GmbH Germany, part of Springer Nature in 2020. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com) and in a subsequent editing, improved by the author. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors.Table of ContentsOption pricing in models in discrete time.- Scorohod's embedding theorem and Donsker's theorem.- Stochastic integration.- Elements of stochastic analysis.- Option pricing in complete and incomplete markets.- Utility optimization, minimum distance martingales, and utility indiff.- Variance-minimum hedging.

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Book SynopsisThis book provides an undergraduate-level introduction to discrete and continuous-time Markov chains and their applications, with a particular focus on the first step analysis technique and its applications to average hitting times and ruin probabilities. It also discusses classical topics such as recurrence and transience, stationary and limiting distributions, as well as branching processes. It first examines in detail two important examples (gambling processes and random walks) before presenting the general theory itself in the subsequent chapters. It also provides an introduction to discrete-time martingales and their relation to ruin probabilities and mean exit times, together with a chapter on spatial Poisson processes. The concepts presented are illustrated by examples, 138 exercises and 9 problems with their solutions.Table of ContentsProbability Background.- Gambling Problems.- Random Walks.- Discrete-Time Markov Chains.- First Step Analysis.- Classification of States.- Long-Run Behavior of Markov Chains.- Branching Processes.- Continuous-Time Markov Chains.- Discrete-Time Martingales.- Spatial Poisson Processes.- Reliability Theory.

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Book SynopsisTrade Review"This textbook contains and describes all the tools one needs to plan and to carry out a simulation study as well as to analyze its results." --J.Wolters, zbMATH Open "It presents the statistics needed to analyze simulated data and to validate the simulation model. In this edition, several new topics are included as well as a number of new exercises." --Vigirdas Mackevicius, zbMATH OpenTable of Contents1. Introduction 2. Elements of Probability 3. Random Numbers 4. Generating Discrete Random Variables 5. Generating Continuous Random Variables 6. The Multivariate Normal Distribution and Copulas 7. The Discrete Event Simulation Approach 8. Statistical Analysis of Simulated Data 9. Variance Reduction Techniques 10. Additional Variance Reduction Techniques 11. Statistical Validation Techniques 12. Markov Chain Monte Carlo Methods

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

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

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Book SynopsisThis popular textbook, now in a revised and expanded third edition, presents a comprehensive course in modern probability theory.Probability plays an increasingly important role not only in mathematics, but also in physics, biology, finance and computer science, helping to understand phenomena such as magnetism, genetic diversity and market volatility, and also to construct efficient algorithms. Starting with the very basics, this textbook covers a wide variety of topics in probability, including many not usually found in introductory books, such as: limit theorems for sums of random variables martingales percolation Markov chains and electrical networks construction of stochastic processes Poisson point process and infinite divisibility large deviation principles and statistical physics Brownian motion stochastic integrals and stochastic differential equations. The presentation is self-contained and mathematically rigorous, with the material on probability theory interspersed with chapters on measure theory to better illustrate the power of abstract concepts.This third edition has been carefully extended and includes new features, such as concise summaries at the end of each section and additional questions to encourage self-reflection, as well as updates to the figures and computer simulations. With a wealth of examples and more than 290 exercises, as well as biographical details of key mathematicians, it will be of use to students and researchers in mathematics, statistics, physics, computer science, economics and biology.Table of Contents1 Basic Measure Theory.- 2 Independence.- 3 Generating Functions.- 4 The Integral.- 5 Moments and Laws of Large Numbers.- 6 Convergence Theorems.- 7 Lp-Spaces and the Radon–Nikodym Theorem.- 8 Conditional Expectations.- 9 Martingales.- 10 Optional Sampling Theorems.- 11 Martingale Convergence Theorems and Their Applications.- 12 Backwards Martingales and Exchangeability.- 13 Convergence of Measures.- 14 Probability Measures on Product Spaces.- 15 Characteristic Functions and the Central Limit Theorem.- 16 Infinitely Divisible Distributions.- 17 Markov Chains.- 18 Convergence of Markov Chains.- 19 Markov Chains and Electrical Networks.- 20 Ergodic Theory.- 21 Brownian Motion.- 22 Law of the Iterated Logarithm.- 23 Large Deviations.- 24 The Poisson Point Process.- 25 The Itô Integral.- 26 Stochastic Differential Equations.- References.- Notation Index.- Name Index.- Subject Index.

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Book SynopsisThis book explores recent topics in quantitative finance with an emphasis on applications and calibration to time-series. This last aspect is often neglected in the existing mathematical finance literature while it is crucial for risk management. The first part of this book focuses on switching regime processes that allow to model economic cycles in financial markets. After a presentation of their mathematical features and applications to stocks and interest rates, the estimation with the Hamilton filter and Markov Chain Monte-Carlo algorithm (MCMC) is detailed. A second part focuses on self-excited processes for modeling the clustering of shocks in financial markets. These processes recently receive a lot of attention from researchers and we focus here on its econometric estimation and its simulation. A chapter is dedicated to estimation of stochastic volatility models. Two chapters are dedicated to the fractional Brownian motion and Gaussian fields. After a summary of their features, we present applications for stock and interest rate modeling. Two chapters focuses on sub-diffusions that allows to replicate illiquidity in financial markets. This book targets undergraduate students who have followed a first course of stochastic finance and practitioners as quantitative analyst or actuaries working in risk management.Trade Review“Hainaut has written a book which in such panorama has a position of its own and which should be considered with great interest. … the book should definitely be considered an excellent and warmly recommended read. It is likely that it will be soon become a reference for those interested in modern topics and for young researchers in particular.” (Gianluca Cassese, zbMATH 1512.91001, 2023)Table of ContentsPreface.- Acknowledgements.- Notations.- 1. Switching Models: Properties and Estimation.- 2. Estimation of Continuous Time Processes by Markov Chain Monte Carlo.- 3. Particle Filtering and Estimation.- 4. Modeling of Spillover Effects in Stock Markets.- 5. Non-Markov Models for Contagion and Spillover.- 6. Fractional Brownian Motion.- 7. Gaussian Fields for Asset Prices.- 8. Lévy Interest Rate Models With a Long Memory.- 9. Affine Volterra Processes and Rough Models.- 10. Sub-Diffusion for Illiquid Markets.- 11. A Fractional Dupire Equation for Jump-Diffusions.- References.

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Book Synopsis​The book The E. M. Stein Lectures on Hardy Spaces is based on a graduate course on real variable Hardy spaces which was given by E.M. Stein at Princeton University in the academic year 1973-1974. Stein, along with C. Fefferman and G. Weiss, pioneered this subject area, removing the theory of Hardy spaces from its traditional dependence on complex variables, and to reveal its real-variable underpinnings.This book is based on Steven G. Krantz’s notes from the course given by Stein. The text builds on Fefferman's theorem that BMO is the dual of the Hardy space. Using maximal functions, singular integrals, and related ideas, Stein offers many new characterizations of the Hardy spaces. The result is a rich tapestry of ideas that develops the theory of singular integrals to a new level. The final chapter describes the major developments since 1974.This monograph is of broad interest to graduate students and researchers in mathematical analysis. Prerequisites for the book include a solid understanding of real variable theory and complex variable theory. A basic knowledge of functional analysis would also be useful.Table of ContentsIntroductory material.- More on Hardy Spaces.- Background on H^p Spaces.- Hardy Spaces on D.- Hardy Spaces on R^n.- Developments Since 1974.- Concluding Remarks.- Bibliography.- Index.

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Book SynopsisThis volume presents recent developments in the area of Lévy-type processes and more general stochastic processes that behave locally like a Lévy process. Although written in a survey style, quite a few results are extensions of known theorems, and others are completely new. The focus is on the symbol of a Lévy-type process: a non-random function which is a counterpart of the characteristic exponent of a Lévy process. The class of stochastic processes which can be associated with a symbol is characterized, various schemes constructing a stochastic process from a given symbol are discussed, and it is shown how one can use the symbol in order to describe the sample path properties of the underlying process. Lastly, the symbol is used to approximate and simulate Levy-type processes.This is the third volume in a subseries of the Lecture Notes in Mathematics called Lévy Matters. Each volume describes a number of important topics in the theory or applications of Lévy processes and pays tribute to the state of the art of this rapidly evolving subject with special emphasis on the non-Brownian world.Table of ContentsA Primer on Feller Semigroups and Feller Processes.- Feller Generators and Symbols.- Construction of Feller Processes.- Transformations of Feller Processes.- Sample Path Properties.- Global Properties.- Approximation.- Open Problems.- References.- Index.

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Book SynopsisIn these lecture notes, we will analyze the behavior of random walk on disordered media by means of both probabilistic and analytic methods, and will study the scaling limits. We will focus on the discrete potential theory and how the theory is effectively used in the analysis of disordered media. The first few chapters of the notes can be used as an introduction to discrete potential theory.Recently, there has been significant progress on the theory of random walk on disordered media such as fractals and random media. Random walk on a percolation cluster(‘the ant in the labyrinth’)is one of the typical examples. In 1986, H. Kesten showed the anomalous behavior of a random walk on a percolation cluster at critical probability. Partly motivated by this work, analysis and diffusion processes on fractals have been developed since the late eighties. As a result, various new methods have been produced to estimate heat kernels on disordered media. These developments are summarized in the notes.Table of ContentsIntroduction.- Weighted graphs and the associated Markov chains.- Heat kernel estimates – General theory.- Heat kernel estimates using effective resistance.- Heat kernel estimates for random weighted graphs.- Alexander-Orbach conjecture holds when two-point functions behave nicely.- Further results for random walk on IIC.- Random conductance model.

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Book SynopsisThis text provides a framework in which the main objectives of the field of uncertainty quantification (UQ) are defined and an overview of the range of mathematical methods by which they can be achieved. Complete with exercises throughout, the book will equip readers with both theoretical understanding and practical experience of the key mathematical and algorithmic tools underlying the treatment of uncertainty in modern applied mathematics. Students and readers alike are encouraged to apply the mathematical methods discussed in this book to their own favorite problems to understand their strengths and weaknesses, also making the text suitable for a self-study.Uncertainty quantification is a topic of increasing practical importance at the intersection of applied mathematics, statistics, computation and numerous application areas in science and engineering. This text is designed as an introduction to UQ for senior undergraduate and graduate students with a mathematical or statistical background and also for researchers from the mathematical sciences or from applications areas who are interested in the field.T. J. Sullivan was Warwick Zeeman Lecturer at the Mathematics Institute of the University of Warwick, United Kingdom, from 2012 to 2015. Since 2015, he is Junior Professor of Applied Mathematics at the Free University of Berlin, Germany, with specialism in Uncertainty and Risk Quantification.Trade Review“Book is one of very few that discuss a vast array of topics in the developing field of uncertainty quantification (UQ). … The text is mathematically rigorous, and though the intended audience is the senior undergraduate or early graduate mathematics student … . this is a book I might recommend to students as a reference for topics related to UQ ... . Overall, this introduction to UQ leaves something to be desired. It is well written … .” (Talea L. Mayo, SIAM Review, Vol. 59 (1), March, 2017)“This book presents a collection of mathematical results related to Uncertainly Quantification (UQ). It is intended as a textbook for senior undergraduate or graduate students with a background in mathematics and statistics. … The book might be suitable for a research seminar where students are exposed for the first time to the mathematics behind UQ.” (Elisabeth Ullmann, Mathematical Reviews, February, 2017)“This book aims to provide an introduction to the mathematics of the quantification of uncertainty. It is intended for students in mathematics and statistics. In the US this would be a graduate level textbook.” (William J. Satzer, MAA Reviews, maa.org, February, 2016)Table of ContentsIntroduction.- Measure and Probability Theory.- Banach and Hilbert Spaces.- Optimization Theory.- Measures of Information and Uncertainty.- Bayesian Inverse Problems.- Filtering and Data Assimilation.- Orthogonal Polynomials and Applications.- Numerical Integration.- Sensitivity Analysis and Model Reduction.- Spectral Expansions.- Stochastic Galerkin Methods.- Non-Intrusive Methods.- Distributional Uncertainty.- References.- Index.

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Book SynopsisThis book covers the classical theory of Markov chains on general state-spaces as well as many recent developments. The theoretical results are illustrated by simple examples, many of which are taken from Markov Chain Monte Carlo methods. The book is self-contained, while all the results are carefully and concisely proven. Bibliographical notes are added at the end of each chapter to provide an overview of the literature. Part I lays the foundations of the theory of Markov chain on general states-space. Part II covers the basic theory of irreducible Markov chains on general states-space, relying heavily on regeneration techniques. These two parts can serve as a text on general state-space applied Markov chain theory. Although the choice of topics is quite different from what is usually covered, where most of the emphasis is put on countable state space, a graduate student should be able to read almost all these developments without any mathematical background deeper than that needed to study countable state space (very little measure theory is required). Part III covers advanced topics on the theory of irreducible Markov chains. The emphasis is on geometric and subgeometric convergence rates and also on computable bounds. Some results appeared for a first time in a book and others are original. Part IV are selected topics on Markov chains, covering mostly hot recent developments.Table of ContentsPart I Foundations.- Markov Chains: Basic Definitions.- Examples of Markov Chains.- Stopping Times and the Strong Markov Property.- Martingales, Harmonic Functions and Polsson-Dirichlet Problems.- Ergodic Theory for Markov Chains.- Part II Irreducible Chains: Basics.- Atomic Chains.- Markov Chains on a Discrete State Space.- Convergence of Atomic Markov Chains.- Small Sets, Irreducibility and Aperiodicity.- Transience, Recurrence and Harris Recurrence.- Splitting Construction and Invariant Measures.- Feller and T-kernels.- Part III Irreducible Chains: Advanced Topics.- Rates of Convergence for Atomic Markov Chains.- Geometric Recurrence and Regularity.- Geometric Rates of Convergence.- (f, r)-recurrence and Regularity.- Subgeometric Rates of Convergence.- Uniform and V-geometric Ergodicity by Operator Methods.- Coupling for Irreducible Kernels.- Part IV Selected Topics.- Convergence in the Wasserstein Distance.- Central Limit Theorems.- Spectral Theory.- Concentration Inequalities.- Appendices.- A Notations.- B Topology, Measure, and Probability.- C Weak Convergence.- D Total and V-total Variation Distances.- E Martingales.- F Mixing Coefficients.- G Solutions to Selected Exercises.

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Book SynopsisThis edition contains detailed solutions of selected exercises. Many readers have requested this, because it makes the book more suitable for self-study. At the same time new exercises (without solutions) have beed added. They have all been placed in the end of each chapter, in order to facilitate the use of this edition together with previous ones. Several errors have been corrected and formulations have been improved. This has been made possible by the valuable comments from (in alphabetical order) Jon Bohlin, Mark Davis, Helge Holden, Patrick Jaillet, Chen Jing, Natalia Koroleva,MarioLefebvre,Alexander Matasov,Thilo Meyer-Brandis, Keigo Osawa, Bjorn Thunestvedt, Jan Uboe and Yngve Williassen. I thank them all for helping to improve the book. My thanks also go to Dina Haraldsson, who once again has performed the typing and drawn the ?gures with great skill. Blindern, September 2002 Bernt Oksendal xv Preface to Corrected Printing, Fifth Edition The main corrections and improvements in this corrected printing are from Chapter 12. I have bene?tted from useful comments from a number of p- ple, including (in alphabetical order) Fredrik Dahl, Simone Deparis, Ulrich Haussmann, Yaozhong Hu, Marianne Huebner, Carl Peter Kirkebo, Ni- lay Kolev, Takashi Kumagai, Shlomo Levental, Geir Magnussen, Anders Oksendal, Jur . . gen Pottho?, Colin Rowat, Stig Sandnes, Lones Smith, S- suo Taniguchi and Bjorn Thunestvedt. I want to thank them all for helping me making the book better. I also want to thank Dina Haraldsson for pro?cient typing.Trade ReviewFrom the reviews of the fifth edition: "This is a highly readable and refreshingly rigorous introduction to stochastic calculus. … This is not a watered-down treatment. It is a serious introduction that starts with fundamental measure-theoretic concepts and ends, coincidentally, with the Black-Scholes formula as one of several examples of applications. This is the best single resource for learning the stochastic calculus … ." (riskbook.com, 2002) From the reviews of the sixth edition: "The book … has evolved from a 200-page typewritten booklet to a modern classic. Part of its charm and success is the fact that the author does not bother too much with the (for the novice) cumbersome rigorous theory … . This does not mean that the book is not rigorous, it is just the timing and dosage of mathematical rigour … that is palatable for undergraduates … . a highly readable account, suitable for self-study and for use in the classroom." (René L. Schilling, The Mathematical Gazette, March, 2005) "This is the sixth edition of the classical and excellent book on stochastic differential equations. The main difference with the next to last edition is the addition of detailed solutions of selected exercises … . This is certainly an excellent idea in view to test its ability of applications of the concepts … . certainly one of the best books on the subject, it will be very helpful to any graduate students and also very valuable for any analysts of financial market." (Stéphane Métens, Physicalia, Vol. 26 (1), 2004) "This is now the sixth edition of the excellent book on stochastic differential equations and related topics. … the presentation is successfully balanced between being easily accessible for a broad audience and being mathematically rigorous. The book is a first choice for courses at graduate level in applied stochastic differential equations. The inclusion of detailed solutions to many of the exercises in this edition also makes it very useful for self-study." (Evelyn Buckwar, Zentralblatt MATH, Vol. 1025, 2003)Table of ContentsSome Mathematical Preliminaries.- Itô Integrals.- The Itô Formula and the Martingale Representation Theorem.- Stochastic Differential Equations.- The Filtering Problem.- Diffusions: Basic Properties.- Other Topics in Diffusion Theory.- Applications to Boundary Value Problems.- Application to Optimal Stopping.- Application to Stochastic Control.- Application to Mathematical Finance.

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Book SynopsisThis introduction can be used, at the beginning graduate level, for a one-semester course on probability theory or for self-direction without benefit of a formal course; the measure theory needed is developed in the text. It will also be useful for students and teachers in related areas such as finance theory, electrical engineering, and operations research. The text covers the essentials in a directed and lean way with 28 short chapters, and assumes only an undergraduate background in mathematics. Readers are taken right up to a knowledge of the basics of Martingale Theory, and the interested student will be ready to continue with the study of more advanced topics, such as Brownian Motion and Ito Calculus, or Statistical Inference.Trade Review"(The book is) a lean and largely self-contained introduction to the modern theory of probability, aimed at advanced undergraduate or beginning graduate students. The 28 short chapters belie the book's genesis as polished lecture notes; the exposition is sleek and rigorous and each chapter ends with a supporting collection of mainly routine exercises. ... The authors make it clear what luggage is required for this exhilarating trek,... a good knowledge of advanced calculus, some linear algebra, and some "mathematical sophistication". With this understood, the itinerary is immaculately paced and planned with just the right balances of technical ascents and pauses to admire the scenery. Within the constraints of a slim volume, it is hard to imagine how the authors could have done a more effective or more attractive job." The Mathematical Gazette, Vol. 84, No 500, 2000 "The authors provide the shortest path through the twenty-eight chapter headings. The topics are treated in a mathematically and pedagogically digestible way. The writing is concise and crisp: the average chapter length is about eight pages. ... Numerous exercises add to the value of the text as a teaching tool. In conclusion, this is an excellent text for the intended audience."Short Book Reviews, Vol. 21, No. 2, 2001Table of Contents1. Introduction 2. Axioms of Probability 3. Conditional Probability and Independence 4. Probabilities on a Countable Space 5. Random Variables on a Countable Space 6. Construction of a Probability Measure 7. Construction of a Probability Measure on R 8. Random Variables 9. Integration with Respect to a Probability Measure 10. Independent Random Variables 11. Probability Distributions on R 12. Probability Distributions on Rn 13. Characteristic Functions 14. Properties of Characteristic Functions 15. Sums of Independent Random Variables 16. Gaussian Random Variables (The Normal and the Multivariate Normal Distributions) 17. Convergence of Random Variables 18. Weak Convergence 19. Weak Convergence and Characteristic Functions 20. The Laws of Large Numbers 21. The Central Limit Theorem 22. L2 and Hilbert Spaces 23. Conditional Expectation 24. Martingales 25. Supermartingales and Submartingales 26. Martingale Inequalities 27. Martingales Convergence Theorems 28. The Radon-Nikodym Theorem

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

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Book SynopsisThis book offers a systematic introduction to the optimal stochastic control theory via the dynamic programming principle, which is a powerful tool to analyze control problems.First we consider completely observable control problems with finite horizons. Using a time discretization we construct a nonlinear semigroup related to the dynamic programming principle (DPP), whose generator provides the Hamilton–Jacobi–Bellman (HJB) equation, and we characterize the value function via the nonlinear semigroup, besides the viscosity solution theory. When we control not only the dynamics of a system but also the terminal time of its evolution, control-stopping problems arise. This problem is treated in the same frameworks, via the nonlinear semigroup. Its results are applicable to the American option price problem.Zero-sum two-player time-homogeneous stochastic differential games and viscosity solutions of the Isaacs equations arising from such games are studied via a nonlinear semigroup related to DPP (the min-max principle, to be precise). Using semi-discretization arguments, we construct the nonlinear semigroups whose generators provide lower and upper Isaacs equations.Concerning partially observable control problems, we refer to stochastic parabolic equations driven by colored Wiener noises, in particular, the Zakai equation. The existence and uniqueness of solutions and regularities as well as Itô's formula are stated. A control problem for the Zakai equations has a nonlinear semigroup whose generator provides the HJB equation on a Banach space. The value function turns out to be a unique viscosity solution for the HJB equation under mild conditions.This edition provides a more generalized treatment of the topic than does the earlier book Lectures on Stochastic Control Theory (ISI Lecture Notes 9), where time-homogeneous cases are dealt with. Here, for finite time-horizon control problems, DPP was formulated as a one-parameter nonlinear semigroup, whose generator provides the HJB equation, by using a time-discretization method. The semigroup corresponds to the value function and is characterized as the envelope of Markovian transition semigroups of responses for constant control processes. Besides finite time-horizon controls, the book discusses control-stopping problems in the same frameworks.

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Book SynopsisThis study provides innovative mathematical models for assessing the eruption probability and associated volcanic hazards, and applies them to the Campi Flegrei caldera in Italy. Throughout the book, significant attention is devoted to quantifying the sources of uncertainty affecting the forecast estimates. The Campi Flegrei caldera is certainly one of the world’s highest-risk volcanoes, with more than 70 eruptions over the last 15,000 years, prevalently explosive ones of varying magnitude, intensity and vent location. In the second half of the twentieth century the volcano apparently once again entered a phase of unrest that continues to the present. Hundreds of thousands of people live inside the caldera and over a million more in the nearby city of Naples, making a future eruption of Campi Flegrei an event with potentially catastrophic consequences at the national and European levels.Table of ContentsIntroduction.- Vent opening probability maps.- Pyroclastic density current invasion maps.- Time-space model for the next eruption.- Addendum.- Supporting information.

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Book SynopsisMost branches of science involving random fluctuations can be approached by Stochastic Calculus. These include, but are not limited to, signal processing, noise filtering, stochastic control, optimal stopping, electrical circuits, financial markets, molecular chemistry, population dynamics, etc. All these applications assume a strong mathematical background, which in general takes a long time to develop. Stochastic Calculus is not an easy to grasp theory, and in general, requires acquaintance with the probability, analysis and measure theory.The goal of this book is to present Stochastic Calculus at an introductory level and not at its maximum mathematical detail. The author's goal was to capture as much as possible the spirit of elementary deterministic Calculus, at which students have been already exposed. This assumes a presentation that mimics similar properties of deterministic Calculus, which facilitates understanding of more complicated topics of Stochastic Calculus.The second edition contains several new features that improved the first edition both qualitatively and quantitatively. First, two more chapters have been added, Chapter 12 and Chapter 13, dealing with applications of stochastic processes in Electrochemistry and global optimization methods.This edition contains also a final chapter material containing fully solved review problems and provides solutions, or at least valuable hints, to all proposed problems. The present edition contains a total of about 250 exercises.This edition has also improved presentation from the first edition in several chapters, including new material.

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

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Book SynopsisThis volume presents topics in probability theory covered during a first-year graduate course given at the Courant Institute of Mathematical Sciences, USA. The necessary background material in measure theory is developed, including all the standard topics.Table of ContentsMeasure theory; Weak convergence; Independent sums; Dependent random variables; Martingales; Stationary stochastic processes; Dynamic programming and filtering; Bibliography; Index

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Book SynopsisProvides a quick, but very readable introduction to stochastic differential equations—that is, to differential equations subject to additive “white noise" and related random disturbances. The exposition is strongly focused upon the interplay between probabilistic intuition and mathematical rigour.Trade Review... [A]n interesting and unusual introduction to stochastic differential equations...topical and appealing to a wide audience. ... This is interesting stuff and, because of Evans' always clear explanations, it is fun too." - MAA ReviewsTable of Contents Preface Introduction A crash course in probability theory Brownian motion and “white noise” Stochastical integrals Stochastic differential equations Applications Appendix Exercises Notes and suggested reading 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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