Probability And Martingales

Probability and Martingales: A Deep Dive into Randomness and Fair Games

Are you fascinated by the unpredictable nature of chance? Do you wonder about the mathematics behind gambling, financial markets, or even the seemingly random events of everyday life? Then you've come to the right place. This comprehensive guide delves into the captivating world of probability and martingales, two interconnected concepts that form the bedrock of much of modern mathematics and its applications. We'll journey from fundamental probability principles to the sophisticated world of martingale theory, exploring its implications in various fields. Prepare to unravel the mysteries of randomness and gain a deeper understanding of how we can model and predict (to a certain extent!) the unpredictable. This guide will equip you with the knowledge to approach probabilistic scenarios with confidence and a new appreciation for the elegance of mathematical reasoning.

I. Understanding the Fundamentals of Probability

Before we embark on the more advanced topic of martingales, let's build a solid foundation in probability theory. Probability, at its core, quantifies the likelihood of an event occurring. We'll explore key concepts such as:

Sample Spaces and Events: Defining the possible outcomes of a random experiment and identifying specific events within that space. We'll cover examples ranging from simple coin flips to more complex scenarios.
Probability Measures: Assigning numerical values (probabilities) to events, adhering to the axioms of probability. We'll delve into different approaches to assigning probabilities, including classical, frequentist, and subjective methods.
Conditional Probability and Bayes' Theorem: Understanding how the probability of an event changes given information about another event. This is crucial for many applications, from medical diagnosis to spam filtering. We'll explore Bayes' Theorem and its practical significance.
Independent Events: Investigating events whose occurrence doesn't influence each other. We'll examine how to determine independence and its impact on calculating joint probabilities.
Discrete and Continuous Random Variables: Distinguishing between variables that can take on only a finite number of values (discrete) and those that can take on any value within a given range (continuous). We'll explore common probability distributions for both types of variables, such as the binomial, Poisson, normal, and exponential distributions.
Expectation and Variance: Calculating the expected value (average) and variance (spread) of random variables, providing crucial measures of central tendency and dispersion.

II. Random Walks and the Intuition Behind Martingales

Martingales are a specific type of stochastic process, which essentially means a sequence of random variables evolving over time. Before diving into the formal definition, let's build intuition by considering random walks. A simple random walk can be visualized as a path taken by someone flipping a coin and moving one step forward for heads and one step backward for tails. This seemingly simple model underpins many complex phenomena.

We'll explore:

The Simple Random Walk: Analyzing its properties, including its expected position and the probability of returning to the origin.
The Gambler's Ruin Problem: A classic problem illustrating the implications of random walks and the possibility of ruin in gambling scenarios.
Building Intuition for Martingales: Introducing the concept of a "fair game" – a game where your expected winnings remain constant regardless of your current fortune. This concept forms the basis of a martingale.

III. Formal Definition and Properties of Martingales

Now we'll delve into the formal mathematical definition of a martingale. This involves concepts from measure theory and stochastic processes, but we'll strive for clarity and accessibility.

We will cover:

The Formal Definition of a Martingale: Presenting the mathematical conditions that define a martingale in terms of conditional expectation.
Types of Martingales: Exploring different types of martingales, including discrete-time and continuous-time martingales.
Key Properties of Martingales: Examining important properties such as the optional stopping theorem, which has significant implications for financial modeling and gambling strategies.
Examples of Martingales: Exploring real-world examples of martingales, including stock prices under certain assumptions and specific gambling games.

IV. Applications of Probability and Martingales

The applications of probability and martingales are vast and far-reaching, spanning diverse fields. We'll touch upon some key areas:

Finance: Modeling stock prices, options pricing (Black-Scholes model), risk management, and portfolio optimization.
Insurance: Actuarial science, risk assessment, and pricing insurance policies.
Gambling: Analyzing gambling strategies, understanding the house edge, and assessing the probabilities of winning.
Statistics: Hypothesis testing, confidence intervals, and Bayesian inference.
Physics: Brownian motion, diffusion processes, and statistical mechanics.

V. Conclusion: Further Exploration of Probability and Martingales

This guide provides a solid foundation in probability and martingales. While we've covered key concepts and applications, the field is vast, and there's much more to explore. We encourage you to continue your learning journey by delving deeper into specific areas that interest you.

Ebook Outline: Probability and Martingales

Title: Mastering Probability and Martingales: From Basics to Advanced Applications

Outline:

Introduction: Hooking the reader and providing an overview of the ebook's content.
Chapter 1: Fundamentals of Probability: Covering basic concepts, probability measures, conditional probability, and common distributions.
Chapter 2: Random Walks and Martingale Intuition: Building intuition through random walks and the gambler's ruin problem.
Chapter 3: Formal Definition and Properties of Martingales: Presenting the mathematical definition, types, and key properties.
Chapter 4: Applications of Probability and Martingales: Exploring applications in finance, insurance, gambling, statistics, and physics.
Conclusion: Summarizing key concepts and suggesting further reading.


(The detailed explanation of each chapter point is provided above in the main article.)

FAQs

1. What is the difference between probability and statistics? Probability deals with the theoretical likelihood of events, while statistics uses data to make inferences about populations.

2. What is a stochastic process? A stochastic process is a sequence of random variables evolving over time.

3. What is the optional stopping theorem? It states that the expected value of a martingale at a stopping time is equal to its initial value, under certain conditions.

4. How are martingales used in finance? They are used to model stock prices (under certain assumptions), price options, and manage risk.

5. What are some common probability distributions? Binomial, Poisson, normal, exponential, and many more.

6. What is a Markov chain? A Markov chain is a specific type of stochastic process where the future depends only on the present state, not the past.

7. What are Brownian motion and its relation to martingales? Brownian motion is a continuous-time stochastic process often used to model random movement; it's closely related to martingales.

8. Can martingales guarantee winning in gambling? No, martingales describe fair games; however, real-world gambling often involves unfair games or limitations.

9. Where can I find more advanced resources on probability and martingales? Look for university-level textbooks on probability theory and stochastic processes.

Related Articles:

1. Introduction to Stochastic Processes: A foundational guide to understanding stochastic processes, setting the stage for martingale theory.

2. The Black-Scholes Model Explained: A detailed look at the famous option pricing model relying heavily on stochastic calculus and martingales.

3. Bayesian Statistics for Beginners: An introduction to Bayesian statistics, which uses probability to update beliefs based on new evidence.

4. Understanding Conditional Probability: A focused exploration of conditional probability and its applications in various fields.

5. Random Walks and Their Applications: A deeper dive into the properties and applications of random walks in different contexts.

6. The Gambler's Ruin Problem: A Mathematical Analysis: A detailed mathematical treatment of the gambler's ruin problem and its implications.

7. Actuarial Science and Risk Management: An overview of actuarial science and how probability and martingales are used for risk assessment.

8. Introduction to Markov Chains: A beginner-friendly explanation of Markov chains and their properties.

9. Stochastic Calculus for Finance: An advanced introduction to stochastic calculus as used in financial modeling.


  probability and martingales: Probability with Martingales David Williams, 1991-02-14 This is a masterly introduction to the modern, and rigorous, theory of probability. The author emphasises martingales and develops all the necessary measure theory.
  probability and martingales: Martingale Limit Theory and Its Application P. Hall, C. C. Heyde, 2014-07-10 Martingale Limit Theory and Its Application discusses the asymptotic properties of martingales, particularly as regards key prototype of probabilistic behavior that has wide applications. The book explains the thesis that martingale theory is central to probability theory, and also examines the relationships between martingales and processes embeddable in or approximated by Brownian motion. The text reviews the martingale convergence theorem, the classical limit theory and analogs, and the martingale limit theorems viewed as the rate of convergence results in the martingale convergence theorem. The book explains the square function inequalities, weak law of large numbers, as well as the strong law of large numbers. The text discusses the reverse martingales, martingale tail sums, the invariance principles in the central limit theorem, and also the law of the iterated logarithm. The book investigates the limit theory for stationary processes via corresponding results for approximating martingales and the estimation of parameters from stochastic processes. The text can be profitably used as a reference for mathematicians, advanced students, and professors of higher mathematics or statistics.
  probability and martingales: Probability Rick Durrett, 2010-08-30 This classic introduction to probability theory for beginning graduate students covers laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems. The fourth edition begins with a short chapter on measure theory to orient readers new to the subject.
  probability and martingales: Measures, Integrals and Martingales René L. Schilling, 2005-11-10 This book, first published in 2005, introduces measure and integration theory as it is needed in many parts of analysis and probability.
  probability and martingales: Probability: A Graduate Course Allan Gut, 2006-03-16 This textbook on the theory of probability starts from the premise that rather than being a purely mathematical discipline, probability theory is an intimate companion of statistics. The book starts with the basic tools, and goes on to cover a number of subjects in detail, including chapters on inequalities, characteristic functions and convergence. This is followed by explanations of the three main subjects in probability: the law of large numbers, the central limit theorem, and the law of the iterated logarithm. After a discussion of generalizations and extensions, the book concludes with an extensive chapter on martingales.
  probability and martingales: Stochastic Calculus and Applications Samuel N. Cohen, Robert J. Elliott, 2015-11-18 Completely 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. Building upon the original release of this title, this text will be of great interest to research mathematicians and graduate students working in those fields, as well as quants in the finance industry. New features of this edition include: End of chapter exercises; New chapters on basic measure theory and Backward SDEs; Reworked proofs, examples and explanatory material; Increased focus on motivating the mathematics; Extensive topical index. Such a self-contained and complete exposition of stochastic calculus and applications fills an existing gap in the literature. The book can be recommended for first-year graduate studies. It will be useful for all who intend to work with stochastic calculus as well as with its applications.–Zentralblatt (from review of the First Edition)
  probability and martingales: An Introduction to Measure and Probability J.C. Taylor, 2012-12-06 Assuming only calculus and linear algebra, Professor Taylor introduces readers to measure theory and probability, discrete martingales, and weak convergence. This is a technically complete, self-contained and rigorous approach that helps the reader to develop basic skills in analysis and probability. Students of pure mathematics and statistics can thus expect to acquire a sound introduction to basic measure theory and probability, while readers with a background in finance, business, or engineering will gain a technical understanding of discrete martingales in the equivalent of one semester. J. C. Taylor is the author of numerous articles on potential theory, both probabilistic and analytic, and is particularly interested in the potential theory of symmetric spaces.
  probability and martingales: Probability Theory Achim Klenke, 2007-12-31 Aimed primarily at graduate students and researchers, this text is a comprehensive course in modern probability theory and its measure-theoretical foundations. It covers a wide variety of topics, many of which are not usually found in introductory textbooks. The theory is developed rigorously and in a self-contained way, with the chapters on measure theory interlaced with the probabilistic chapters in order to display the power of the abstract concepts in the world of probability theory. In addition, plenty of figures, computer simulations, biographic details of key mathematicians, and a wealth of examples support and enliven the presentation.
  probability and martingales: Theory of Martingales Robert Liptser, A.N. Shiryayev, 2012-12-06 One service mathematics has rc:ndered the 'Et moi, ', si j'avait su comment CD revenir, je n'y serais point alle. ' human race. It has put common SCIIJC back Jules Verne where it belongs. on the topmost shelf next to tbe dusty canister 1abdled 'discarded non- The series is divergent; tberefore we may be sense'. able to do sometbing witb it Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ... '; 'One service logic has rendered com puter science ... '; 'One service category theory has rendered mathematics ... '. All arguably true_ And all statements obtainable this way form part of the raison d'etre of this series_ This series, Mathematics and Its ApplicatiOns, started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope_ At the time I wrote Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches.
  probability and martingales: Brownian Motion, Martingales, and Stochastic Calculus Jean-François Le Gall, 2016-04-28 This 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.
  probability and martingales: Knowing the Odds John B. Walsh, 2023-08-16 John Walsh, one of the great masters of the subject, has written a superb book on probability. It covers at a leisurely pace all the important topics that students need to know, and provides excellent examples. I regret his book was not available when I taught such a course myself, a few years ago. —Ioannis Karatzas, Columbia University In this wonderful book, John Walsh presents a panoramic view of Probability Theory, starting from basic facts on mean, median and mode, continuing with an excellent account of Markov chains and martingales, and culminating with Brownian motion. Throughout, the author's personal style is apparent; he manages to combine rigor with an emphasis on the key ideas so the reader never loses sight of the forest by being surrounded by too many trees. As noted in the preface, “To teach a course with pleasure, one should learn at the same time.” Indeed, almost all instructors will learn something new from the book (e.g. the potential-theoretic proof of Skorokhod embedding) and at the same time, it is attractive and approachable for students. —Yuval Peres, Microsoft With many examples in each section that enhance the presentation, this book is a welcome addition to the collection of books that serve the needs of advanced undergraduate as well as first year graduate students. The pace is leisurely which makes it more attractive as a text. —Srinivasa Varadhan, Courant Institute, New York This book covers in a leisurely manner all the standard material that one would want in a full year probability course with a slant towards applications in financial analysis at the graduate or senior undergraduate honors level. It contains a fair amount of measure theory and real analysis built in but it introduces sigma-fields, measure theory, and expectation in an especially elementary and intuitive way. A large variety of examples and exercises in each chapter enrich the presentation in the text.
  probability and martingales: Probability and Stochastics Erhan Çınlar, 2011-02-21 This text is an introduction to the modern theory and applications of probability and stochastics. The style and coverage is geared towards the theory of stochastic processes, but with some attention to the applications. In many instances the gist of the problem is introduced in practical, everyday language and then is made precise in mathematical form. The first four chapters are on probability theory: measure and integration, probability spaces, conditional expectations, and the classical limit theorems. There follows chapters on martingales, Poisson random measures, Levy Processes, Brownian motion, and Markov Processes. Special attention is paid to Poisson random measures and their roles in regulating the excursions of Brownian motion and the jumps of Levy and Markov processes. Each chapter has a large number of varied examples and exercises. The book is based on the author’s lecture notes in courses offered over the years at Princeton University. These courses attracted graduate students from engineering, economics, physics, computer sciences, and mathematics. Erhan Cinlar has received many awards for excellence in teaching, including the President’s Award for Distinguished Teaching at Princeton University. His research interests include theories of Markov processes, point processes, stochastic calculus, and stochastic flows. The book is full of insights and observations that only a lifetime researcher in probability can have, all told in a lucid yet precise style.
  probability and martingales: Continuous Martingales and Brownian Motion Daniel Revuz, Marc Yor, 2013-03-09 This is a magnificent book! Its purpose is to describe in considerable detail a variety of techniques used by probabilists in the investigation of problems concerning Brownian motion....This is THE book for a capable graduate student starting out on research in probability: the effect of working through it is as if the authors are sitting beside one, enthusiastically explaining the theory, presenting further developments as exercises. –BULLETIN OF THE L.M.S.
  probability and martingales: A First Look At Stochastic Processes Jeffrey S Rosenthal, 2019-09-26 This textbook introduces the theory of stochastic processes, that is, randomness which proceeds in time. Using concrete examples like repeated gambling and jumping frogs, it presents fundamental mathematical results through simple, clear, logical theorems and examples. It covers in detail such essential material as Markov chain recurrence criteria, the Markov chain convergence theorem, and optional stopping theorems for martingales. The final chapter provides a brief introduction to Brownian motion, Markov processes in continuous time and space, Poisson processes, and renewal theory.Interspersed throughout are applications to such topics as gambler's ruin probabilities, random walks on graphs, sequence waiting times, branching processes, stock option pricing, and Markov Chain Monte Carlo (MCMC) algorithms.The focus is always on making the theory as well-motivated and accessible as possible, to allow students and readers to learn this fascinating subject as easily and painlessly as possible.
  probability and martingales: Probability Theory and Stochastic Processes Pierre Brémaud, 2020-04-07 The ultimate objective of this book is to present a panoramic view of the main stochastic processes which have an impact on applications, with complete proofs and exercises. Random processes play a central role in the applied sciences, including operations research, insurance, finance, biology, physics, computer and communications networks, and signal processing. In order to help the reader to reach a level of technical autonomy sufficient to understand the presented models, this book includes a reasonable dose of probability theory. On the other hand, the study of stochastic processes gives an opportunity to apply the main theoretical results of probability theory beyond classroom examples and in a non-trivial manner that makes this discipline look more attractive to the applications-oriented student. One can distinguish three parts of this book. The first four chapters are about probability theory, Chapters 5 to 8 concern random sequences, or discrete-time stochastic processes, and the rest of the book focuses on stochastic processes and point processes. There is sufficient modularity for the instructor or the self-teaching reader to design a course or a study program adapted to her/his specific needs. This book is in a large measure self-contained.
  probability and martingales: Martingale Methods in Financial Modelling Marek Musiela, 2013-06-29 A comprehensive and self-contained treatment of the theory and practice of option pricing. The role of martingale methods in financial modeling is exposed. The emphasis is on using arbitrage-free models already accepted by the market as well as on building the new ones. Standard calls and puts together with numerous examples of exotic options such as barriers and quantos, for example on stocks, indices, currencies and interest rates are analysed. The importance of choosing a convenient numeraire in price calculations is explained. Mathematical and financial language is used so as to bring mathematicians closer to practical problems of finance and presenting to the industry useful maths tools.
  probability and martingales: Classical Potential Theory and Its Probabilistic Counterpart Joseph L. Doob, 2001-01-12 From the reviews: Here is a momumental work by Doob, one of the masters, in which Part 1 develops the potential theory associated with Laplace's equation and the heat equation, and Part 2 develops those parts (martingales and Brownian motion) of stochastic process theory which are closely related to Part 1. --G.E.H. Reuter in Short Book Reviews (1985)
  probability and martingales: A Basic Course in Measure and Probability Ross Leadbetter, Stamatis Cambanis, Vladas Pipiras, 2014-01-30 A concise introduction covering all of the measure theory and probability most useful for statisticians.
  probability and martingales: Martingale Methods in Statistics Yoichi Nishiyama, 2021-11-24 Martingale Methods in Statistics provides a unique introduction to statistics of stochastic processes written with the author’s strong desire to present what is not available in other textbooks. While the author chooses to omit the well-known proofs of some of fundamental theorems in martingale theory by making clear citations instead, the author does his best to describe some intuitive interpretations or concrete usages of such theorems. On the other hand, the exposition of relatively new theorems in asymptotic statistics is presented in a completely self-contained way. Some simple, easy-to-understand proofs of martingale central limit theorems are included. The potential readers include those who hope to build up mathematical bases to deal with high-frequency data in mathematical finance and those who hope to learn the theoretical background for Cox’s regression model in survival analysis. A highlight of the monograph is Chapters 8-10 dealing with Z-estimators and related topics, such as the asymptotic representation of Z-estimators, the theory of asymptotically optimal inference based on the LAN concept and the unified approach to the change point problems via Z-process method. Some new inequalities for maxima of finitely many martingales are presented in the Appendix. Readers will find many tips for solving concrete problems in modern statistics of stochastic processes as well as in more fundamental models such as i.i.d. and Markov chain models.
  probability and martingales: Foundations of Modern Probability Olav Kallenberg, 2002-01-08 The first edition of this single volume on the theory of probability has become a highly-praised standard reference for many areas of probability theory. Chapters from the first edition have been revised and corrected, and this edition contains four new chapters. New material covered includes multivariate and ratio ergodic theorems, shift coupling, Palm distributions, Harris recurrence, invariant measures, and strong and weak ergodicity.
  probability and martingales: Set-Indexed Martingales B.G. Ivanoff, Ely Merzbach, 1999-10-27 Set-Indexed Martingales offers a unique, comprehensive development of a general theory of Martingales indexed by a family of sets. The authors establish-for the first time-an appropriate framework that provides a suitable structure for a theory of Martingales with enough generality to include many interesting examples. Developed from first principles, the theory brings together the theories of Martingales with a directed index set and set-indexed stochastic processes. Part One presents several classical concepts extended to this setting, including: stopping, predictability, Doob-Meyer decompositions, martingale characterizations of the set-indexed Poisson process, and Brownian motion. Part Two addresses convergence of sequences of set-indexed processes and introduces functional convergence for processes whose sample paths live in a Skorokhod-type space and semi-functional convergence for processes whose sample paths may be badly behaved. Completely self-contained, the theoretical aspects of this work are rich and promising. With its many important applications-especially in the theory of spatial statistics and in stochastic geometry- Set Indexed Martingales will undoubtedly generate great interest and inspire further research and development of the theory and applications.
  probability and martingales: High-Dimensional Probability Roman Vershynin, 2018-09-27 An integrated package of powerful probabilistic tools and key applications in modern mathematical data science.
  probability and martingales: A User's Guide to Measure Theoretic Probability David Pollard, 2002 This book grew from a one-semester course offered for many years to a mixed audience of graduate and undergraduate students who have not had the luxury of taking a course in measure theory. The core of the book covers the basic topics of independence, conditioning, martingales, convergence in distribution, and Fourier transforms. In addition there are numerous sections treating topics traditionally thought of as more advanced, such as coupling and the KMT strong approximation, option pricing via the equivalent martingale measure, and the isoperimetric inequality for Gaussian processes. The book is not just a presentation of mathematical theory, but is also a discussion of why that theory takes its current form. It will be a secure starting point for anyone who needs to invoke rigorous probabilistic arguments and understand what they mean.
  probability and martingales: Probability and Measure Patrick Billingsley, 2017 Now in its new third edition, Probability and Measure offers advanced students, scientists, and engineers an integrated introduction to measure theory and probability. Retaining the unique approach of the previous editions, this text interweaves material on probability and measure, so that probability problems generate an interest in measure theory and measure theory is then developed and applied to probability. Probability and Measure provides thorough coverage of probability, measure, integration, random variables and expected values, convergence of distributions, derivatives and conditional probability, and stochastic processes. The Third Edition features an improved treatment of Brownian motion and the replacement of queuing theory with ergodic theory.· Probability· Measure· Integration· Random Variables and Expected Values· Convergence of Distributions· Derivatives and Conditional Probability· Stochastic Processes
  probability and martingales: Understanding Probability Henk Tijms, 2007-07-26 In this fully revised second edition of Understanding Probability, the reader can learn about the world of probability in an informal way. The author demystifies the law of large numbers, betting systems, random walks, the bootstrap, rare events, the central limit theorem, the Bayesian approach and more. This second edition has wider coverage, more explanations and examples and exercises, and a new chapter introducing Markov chains, making it a great choice for a first probability course. But its easy-going style makes it just as valuable if you want to learn about the subject on your own, and high school algebra is really all the mathematical background you need.
  probability and martingales: Continuous Exponential Martingales and BMO Norihiko Kazamaki, 2006-11-15 In three chapters on Exponential Martingales, BMO-martingales, and Exponential of BMO, this book explains in detail the beautiful properties of continuous exponential martingales that play an essential role in various questions concerning the absolute continuity of probability laws of stochastic processes. The second and principal aim is to provide a full report on the exciting results on BMO in the theory of exponential martingales. The reader is assumed to be familiar with the general theory of continuous martingales.
  probability and martingales: Introduction to Probability Joseph K. Blitzstein, Jessica Hwang, 2014-07-24 Developed from celebrated Harvard statistics lectures, Introduction to Probability provides essential language and tools for understanding statistics, randomness, and uncertainty. The book explores a wide variety of applications and examples, ranging from coincidences and paradoxes to Google PageRank and Markov chain Monte Carlo (MCMC). Additional
  probability and martingales: Concentration Inequalities for Sums and Martingales Bernard Bercu, Bernard Delyon, Emmanuel Rio, 2015-09-29 The purpose of this book is to provide an overview of historical and recent results on concentration inequalities for sums of independent random variables and for martingales. The first chapter is devoted to classical asymptotic results in probability such as the strong law of large numbers and the central limit theorem. Our goal is to show that it is really interesting to make use of concentration inequalities for sums and martingales. The second chapter deals with classical concentration inequalities for sums of independent random variables such as the famous Hoeffding, Bennett, Bernstein and Talagrand inequalities. Further results and improvements are also provided such as the missing factors in those inequalities. The third chapter concerns concentration inequalities for martingales such as Azuma-Hoeffding, Freedman and De la Pena inequalities. Several extensions are also provided. The fourth chapter is devoted to applications of concentration inequalities in probability and statistics.
  probability and martingales: Probability Albert Shiryaev, 2013-11-11 In the Preface to the first edition, originally published in 1980, we mentioned that this book was based on the author's lectures in the Department of Mechanics and Mathematics of the Lomonosov University in Moscow, which were issued, in part, in mimeographed form under the title Probabil ity, Statistics, and Stochastic Processors, I, II and published by that Univer sity. Our original intention in writing the first edition of this book was to divide the contents into three parts: probability, mathematical statistics, and theory of stochastic processes, which corresponds to an outline of a three semester course of lectures for university students of mathematics. However, in the course of preparing the book, it turned out to be impossible to realize this intention completely, since a full exposition would have required too much space. In this connection, we stated in the Preface to the first edition that only probability theory and the theory of random processes with discrete time were really adequately presented. Essentially all of the first edition is reproduced in this second edition. Changes and corrections are, as a rule, editorial, taking into account com ments made by both Russian and foreign readers of the Russian original and ofthe English and Germantranslations [Sll]. The author is grateful to all of these readers for their attention, advice, and helpful criticisms. In this second English edition, new material also has been added, as follows: in Chapter 111, §5, §§7-12; in Chapter IV, §5; in Chapter VII, §§8-10.
  probability and martingales: A Course in Probability Theory Kai Lai Chung, 2014-06-28 This book contains about 500 exercises consisting mostly of special cases and examples, second thoughts and alternative arguments, natural extensions, and some novel departures. With a few obvious exceptions they are neither profound nor trivial, and hints and comments are appended to many of them. If they tend to be somewhat inbred, at least they are relevant to the text and should help in its digestion. As a bold venture I have marked a few of them with a * to indicate a must, although no rigid standard of selection has been used. Some of these are needed in the book, but in any case the reader's study of the text will be more complete after he has tried at least those problems.
  probability and martingales: Foundations of Constructive Probability Theory Yuen-Kwok Chan, 2021-05-27 This book provides a systematic and general theory of probability within the framework of constructive mathematics.
  probability and martingales: Probabilities and Potential, B C. Dellacherie, P.-A. Meyer, 2011-08-18 Probabilities and Potential, B
  probability and martingales: Measure, Integral and Probability Marek Capinski, (Peter) Ekkehard Kopp, 2013-06-29 This very well written and accessible book emphasizes the reasons for studying measure theory, which is the foundation of much of probability. By focusing on measure, many illustrative examples and applications, including a thorough discussion of standard probability distributions and densities, are opened. The book also includes many problems and their fully worked solutions.
  probability and martingales: Hilbert Space Methods in Probability and Statistical Inference Christopher G. Small, Don L. McLeish, 2011-09-15 Explains how Hilbert space techniques cross the boundaries into the foundations of probability and statistics. Focuses on the theory of martingales stochastic integration, interpolation and density estimation. Includes a copious amount of problems and examples.
  probability and martingales: Probability for Statisticians Galen R. Shorack, 2006-05-02 The choice of examples used in this text clearly illustrate its use for a one-year graduate course. The material to be presented in the classroom constitutes a little more than half the text, while the rest of the text provides background, offers different routes that could be pursued in the classroom, as well as additional material that is appropriate for self-study. Of particular interest is a presentation of the major central limit theorems via Steins method either prior to or alternative to a characteristic function presentation. Additionally, there is considerable emphasis placed on the quantile function as well as the distribution function, with both the bootstrap and trimming presented. The section on martingales covers censored data martingales.
  probability and martingales: Probability Essentials Jean Jacod, Philip Protter, 2012-12-06 This 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.
  probability and martingales: A Modern Approach to Probability Theory Bert E. Fristedt, Lawrence F. Gray, 2013-11-21 Students and teachers of mathematics and related fields will find this book a comprehensive and modern approach to probability theory, providing the background and techniques to go from the beginning graduate level to the point of specialization in research areas of current interest. The book is designed for a two- or three-semester course, assuming only courses in undergraduate real analysis or rigorous advanced calculus, and some elementary linear algebra. A variety of applications—Bayesian statistics, financial mathematics, information theory, tomography, and signal processing—appear as threads to both enhance the understanding of the relevant mathematics and motivate students whose main interests are outside of pure areas.
  probability and martingales: Discrete Stochastic Processes Robert G. Gallager, 2012-12-06 Stochastic processes are found in probabilistic systems that evolve with time. Discrete stochastic processes change by only integer time steps (for some time scale), or are characterized by discrete occurrences at arbitrary times. Discrete Stochastic Processes helps the reader develop the understanding and intuition necessary to apply stochastic process theory in engineering, science and operations research. The book approaches the subject via many simple examples which build insight into the structure of stochastic processes and the general effect of these phenomena in real systems. The book presents mathematical ideas without recourse to measure theory, using only minimal mathematical analysis. In the proofs and explanations, clarity is favored over formal rigor, and simplicity over generality. Numerous examples are given to show how results fail to hold when all the conditions are not satisfied. Audience: An excellent textbook for a graduate level course in engineering and operations research. Also an invaluable reference for all those requiring a deeper understanding of the subject.
  probability and martingales: Probability Theory S. R. S. Varadhan, 2001-09-10 This volume presents topics in probability theory covered during a first-year graduate course given at the Courant Institute of Mathematical Sciences. The necessary background material in measure theory is developed, including the standard topics, such as extension theorem, construction of measures, integration, product spaces, Radon-Nikodym theorem, and conditional expectation. In the first part of the book, characteristic functions are introduced, followed by the study of weak convergence of probability distributions. Then both the weak and strong limit theorems for sums of independent random variables are proved, including the weak and strong laws of large numbers, central limit theorems, laws of the iterated logarithm, and the Kolmogorov three series theorem. The first part concludes with infinitely divisible distributions and limit theorems for sums of uniformly infinitesimal independent random variables. The second part of the book mainly deals with dependent random variables, particularly martingales and Markov chains. Topics include standard results regarding discrete parameter martingales and Doob's inequalities. The standard topics in Markov chains are treated, i.e., transience, and null and positive recurrence. A varied collection of examples is given to demonstrate the connection between martingales and Markov chains. Additional topics covered in the book include stationary Gaussian processes, ergodic theorems, dynamic programming, optimal stopping, and filtering. A large number of examples and exercises is included. The book is a suitable text for a first-year graduate course in probability.
  probability and martingales: Radically Elementary Probability Theory Edward Nelson, 1987 Using only the very elementary framework of finite probability spaces, this book treats a number of topics in the modern theory of stochastic processes. This is made possible by using a small amount of Abraham Robinson's nonstandard analysis and not attempting to convert the results into conventional form.