Random Walk Brownian Motion And Martingales

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Random Walk, Brownian Motion, and Martingales

Author : Rabi Bhattacharya
Publisher : Springer Nature
ISBN 13 : 303078939X
Total Pages : 396 pages
Book Rating : 4.0/5 (37 download)

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Book Synopsis Random Walk, Brownian Motion, and Martingales by : Rabi Bhattacharya

Download or read book Random Walk, Brownian Motion, and Martingales written by Rabi Bhattacharya and published by Springer Nature. This book was released on 2021-09-20 with total page 396 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook offers an approachable introduction to stochastic processes that explores the four pillars of random walk, branching processes, Brownian motion, and martingales. Building from simple examples, the authors focus on developing context and intuition before formalizing the theory of each topic. This inviting approach illuminates the key ideas and computations in the proofs, forming an ideal basis for further study. Consisting of many short chapters, the book begins with a comprehensive account of the simple random walk in one dimension. From here, different paths may be chosen according to interest. Themes span Poisson processes, branching processes, the Kolmogorov–Chentsov theorem, martingales, renewal theory, and Brownian motion. Special topics follow, showcasing a selection of important contemporary applications, including mathematical finance, optimal stopping, ruin theory, branching random walk, and equations of fluids. Engaging exercises accompany the theory throughout. Random Walk, Brownian Motion, and Martingales is an ideal introduction to the rigorous study of stochastic processes. Students and instructors alike will appreciate the accessible, example-driven approach. A single, graduate-level course in probability is assumed.

Random Walk, Brownian Motion, and Martingales

Author : Rabi Bhattacharya
Publisher :
ISBN 13 : 9783030789381
Total Pages : 0 pages
Book Rating : 4.7/5 (893 download)

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Book Synopsis Random Walk, Brownian Motion, and Martingales by : Rabi Bhattacharya

Download or read book Random Walk, Brownian Motion, and Martingales written by Rabi Bhattacharya and published by . This book was released on 2021 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook offers an approachable introduction to stochastic processes that explores the four pillars of random walk, branching processes, Brownian motion, and martingales. Building from simple examples, the authors focus on developing context and intuition before formalizing the theory of each topic. This inviting approach illuminates the key ideas and computations in the proofs, forming an ideal basis for further study. Consisting of many short chapters, the book begins with a comprehensive account of the simple random walk in one dimension. From here, different paths may be chosen according to interest. Themes span Poisson processes, branching processes, the Kolmogorov-Chentsov theorem, martingales, renewal theory, and Brownian motion. Special topics follow, showcasing a selection of important contemporary applications, including mathematical finance, optimal stopping, ruin theory, branching random walk, and equations of fluids. Engaging exercises accompany the theory throughout. Random Walk, Brownian Motion, and Martingales is an ideal introduction to the rigorous study of stochastic processes. Students and instructors alike will appreciate the accessible, example-driven approach. A single, graduate-level course in probability is assumed.

Brownian Motion

Author : Peter Mörters
Publisher : Cambridge University Press
ISBN 13 : 1139486578
Total Pages : pages
Book Rating : 4.1/5 (394 download)

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Book Synopsis Brownian Motion by : Peter Mörters

Download or read book Brownian Motion written by Peter Mörters and published by Cambridge University Press. This book was released on 2010-03-25 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This eagerly awaited textbook covers everything the graduate student in probability wants to know about Brownian motion, as well as the latest research in the area. Starting with the construction of Brownian motion, the book then proceeds to sample path properties like continuity and nowhere differentiability. Notions of fractal dimension are introduced early and are used throughout the book to describe fine properties of Brownian paths. The relation of Brownian motion and random walk is explored from several viewpoints, including a development of the theory of Brownian local times from random walk embeddings. Stochastic integration is introduced as a tool and an accessible treatment of the potential theory of Brownian motion clears the path for an extensive treatment of intersections of Brownian paths. An investigation of exceptional points on the Brownian path and an appendix on SLE processes, by Oded Schramm and Wendelin Werner, lead directly to recent research themes.

Intersections of Random Walks

Author : Gregory F. Lawler
Publisher : Springer Science & Business Media
ISBN 13 : 1461459729
Total Pages : 226 pages
Book Rating : 4.4/5 (614 download)

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Book Synopsis Intersections of Random Walks by : Gregory F. Lawler

Download or read book Intersections of Random Walks written by Gregory F. Lawler and published by Springer Science & Business Media. This book was released on 2012-11-06 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: A central study in Probability Theory is the behavior of fluctuation phenomena of partial sums of different types of random variable. One of the most useful concepts for this purpose is that of the random walk which has applications in many areas, particularly in statistical physics and statistical chemistry. Originally published in 1991, Intersections of Random Walks focuses on and explores a number of problems dealing primarily with the nonintersection of random walks and the self-avoiding walk. Many of these problems arise in studying statistical physics and other critical phenomena. Topics include: discrete harmonic measure, including an introduction to diffusion limited aggregation (DLA); the probability that independent random walks do not intersect; and properties of walks without self-intersections. The present softcover reprint includes corrections and addenda from the 1996 printing, and makes this classic monograph available to a wider audience. With a self-contained introduction to the properties of simple random walks, and an emphasis on rigorous results, the book will be useful to researchers in probability and statistical physics and to graduate students interested in basic properties of random walks.

Introduction to Stochastic Calculus with Applications

Author : Fima C. Klebaner
Publisher : Imperial College Press
ISBN 13 : 1860945554
Total Pages : 431 pages
Book Rating : 4.8/5 (69 download)

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Book Synopsis Introduction to Stochastic Calculus with Applications by : Fima C. Klebaner

Download or read book Introduction to Stochastic Calculus with Applications written by Fima C. Klebaner and published by Imperial College Press. This book was released on 2005 with total page 431 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a concise treatment of stochastic calculus and its applications. It gives a simple but rigorous treatment of the subject including a range of advanced topics, it is useful for practitioners who use advanced theoretical results. It covers advanced applications, such as models in mathematical finance, biology and engineering.Self-contained and unified in presentation, the book contains many solved examples and exercises. It may be used as a textbook by advanced undergraduates and graduate students in stochastic calculus and financial mathematics. It is also suitable for practitioners who wish to gain an understanding or working knowledge of the subject. For mathematicians, this book could be a first text on stochastic calculus; it is good companion to more advanced texts by a way of examples and exercises. For people from other fields, it provides a way to gain a working knowledge of stochastic calculus. It shows all readers the applications of stochastic calculus methods and takes readers to the technical level required in research and sophisticated modelling.This second edition contains a new chapter on bonds, interest rates and their options. New materials include more worked out examples in all chapters, best estimators, more results on change of time, change of measure, random measures, new results on exotic options, FX options, stochastic and implied volatility, models of the age-dependent branching process and the stochastic Lotka-Volterra model in biology, non-linear filtering in engineering and five new figures.Instructors can obtain slides of the text from the author.

Random Walk and the Heat Equation

Author : Gregory F. Lawler
Publisher : American Mathematical Soc.
ISBN 13 : 0821848291
Total Pages : 170 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Random Walk and the Heat Equation by : Gregory F. Lawler

Download or read book Random Walk and the Heat Equation written by Gregory F. Lawler and published by American Mathematical Soc.. This book was released on 2010-11-22 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: The heat equation can be derived by averaging over a very large number of particles. Traditionally, the resulting PDE is studied as a deterministic equation, an approach that has brought many significant results and a deep understanding of the equation and its solutions. By studying the heat equation and considering the individual random particles, however, one gains further intuition into the problem. While this is now standard for many researchers, this approach is generally not presented at the undergraduate level. In this book, Lawler introduces the heat equations and the closely related notion of harmonic functions from a probabilistic perspective. The theme of the first two chapters of the book is the relationship between random walks and the heat equation. This first chapter discusses the discrete case, random walk and the heat equation on the integer lattice; and the second chapter discusses the continuous case, Brownian motion and the usual heat equation. Relationships are shown between the two. For example, solving the heat equation in the discrete setting becomes a problem of diagonalization of symmetric matrices, which becomes a problem in Fourier series in the continuous case. Random walk and Brownian motion are introduced and developed from first principles. The latter two chapters discuss different topics: martingales and fractal dimension, with the chapters tied together by one example, a random Cantor set. The idea of this book is to merge probabilistic and deterministic approaches to heat flow. It is also intended as a bridge from undergraduate analysis to graduate and research perspectives. The book is suitable for advanced undergraduates, particularly those considering graduate work in mathematics or related areas.

Probability

Author : Rick Durrett
Publisher : Cambridge University Press
ISBN 13 : 113949113X
Total Pages : pages
Book Rating : 4.1/5 (394 download)

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Book Synopsis Probability by : Rick Durrett

Download or read book Probability written by Rick Durrett and published by Cambridge University Press. This book was released on 2010-08-30 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Continuous Martingales and Brownian Motion

Author : Daniel Revuz
Publisher : Springer Science & Business Media
ISBN 13 : 3662217260
Total Pages : 544 pages
Book Rating : 4.6/5 (622 download)

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Book Synopsis Continuous Martingales and Brownian Motion by : Daniel Revuz

Download or read book Continuous Martingales and Brownian Motion written by Daniel Revuz and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 544 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on the probabilistic theory ofBrownian motion. This is a good topic to center a discussion around because Brownian motion is in the intersec tioll of many fundamental classes of processes. It is a continuous martingale, a Gaussian process, a Markov process or more specifically a process with in dependent increments; it can actually be defined, up to simple transformations, as the real-valued, centered process with independent increments and continuous paths. It is therefore no surprise that a vast array of techniques may be success fully applied to its study and we, consequently, chose to organize the book in the following way. After a first chapter where Brownian motion is introduced, each of the following ones is devoted to a new technique or notion and to some of its applications to Brownian motion. Among these techniques, two are of para mount importance: stochastic calculus, the use ofwhich pervades the whole book and the powerful excursion theory, both of which are introduced in a self contained fashion and with a minimum of apparatus. They have made much easier the proofs of many results found in the epoch-making book of Itö and McKean: Diffusion Processes and their Sampie Paths, Springer (1965).

Brownian Motion

Author : René L. Schilling
Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110307308
Total Pages : 424 pages
Book Rating : 4.1/5 (13 download)

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Book Synopsis Brownian Motion by : René L. Schilling

Download or read book Brownian Motion written by René L. Schilling and published by Walter de Gruyter GmbH & Co KG. This book was released on 2014-06-18 with total page 424 pages. Available in PDF, EPUB and Kindle. Book excerpt: Brownian motion is one of the most important stochastic processes in continuous time and with continuous state space. Within the realm of stochastic processes, Brownian motion is at the intersection of Gaussian processes, martingales, Markov processes, diffusions and random fractals, and it has influenced the study of these topics. Its central position within mathematics is matched by numerous applications in science, engineering and mathematical finance. Often textbooks on probability theory cover, if at all, Brownian motion only briefly. On the other hand, there is a considerable gap to more specialized texts on Brownian motion which is not so easy to overcome for the novice. The authors’ aim was to write a book which can be used as an introduction to Brownian motion and stochastic calculus, and as a first course in continuous-time and continuous-state Markov processes. They also wanted to have a text which would be both a readily accessible mathematical back-up for contemporary applications (such as mathematical finance) and a foundation to get easy access to advanced monographs. This textbook, tailored to the needs of graduate and advanced undergraduate students, covers Brownian motion, starting from its elementary properties, certain distributional aspects, path properties, and leading to stochastic calculus based on Brownian motion. It also includes numerical recipes for the simulation of Brownian motion.

Continuous Martingales and Brownian Motion

Author : D. Revuz
Publisher : Springer
ISBN 13 :
Total Pages : 560 pages
Book Rating : 4.:/5 (318 download)

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Book Synopsis Continuous Martingales and Brownian Motion by : D. Revuz

Download or read book Continuous Martingales and Brownian Motion written by D. Revuz and published by Springer. This book was released on 1991-01-09 with total page 560 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on the probabilistic theory ofBrownian motion. This is a good topic to center a discussion around because Brownian motion is in the intersec tioll of many fundamental classes of processes. It is a continuous martingale, a Gaussian process, a Markov process or more specifically a process with in dependent increments; it can actually be defined, up to simple transformations, as the real-valued, centered process with independent increments and continuous paths. It is therefore no surprise that a vast array of techniques may be success fully applied to its study and we, consequently, chose to organize the book in the following way. After a first chapter where Brownian motion is introduced, each of the following ones is devoted to a new technique or notion and to some of its applications to Brownian motion. Among these techniques, two are of para mount importance: stochastic calculus, the use ofwhich pervades the whole book and the powerful excursion theory, both of which are introduced in a self contained fashion and with a minimum of apparatus. They have made much easier the proofs of many results found in the epoch-making book of Itö and McKean: Diffusion Processes and their Sampie Paths, Springer (1965).

Intersections of Random Walks

Author : Gregory F. Lawler
Publisher : Springer Science & Business Media
ISBN 13 : 1475721374
Total Pages : 219 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis Intersections of Random Walks by : Gregory F. Lawler

Download or read book Intersections of Random Walks written by Gregory F. Lawler and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 219 pages. Available in PDF, EPUB and Kindle. Book excerpt: A more accurate title for this book would be "Problems dealing with the non-intersection of paths of random walks. " These include: harmonic measure, which can be considered as a problem of nonintersection of a random walk with a fixed set; the probability that the paths of independent random walks do not intersect; and self-avoiding walks, i. e. , random walks which have no self-intersections. The prerequisite is a standard measure theoretic course in probability including martingales and Brownian motion. The first chapter develops the facts about simple random walk that will be needed. The discussion is self-contained although some previous expo sure to random walks would be helpful. Many of the results are standard, and I have made borrowed from a number of sources, especially the ex cellent book of Spitzer [65]. For the sake of simplicity I have restricted the discussion to simple random walk. Of course, many of the results hold equally well for more general walks. For example, the local central limit theorem can be proved for any random walk whose increments have mean zero and finite variance. Some of the later results, especially in Section 1. 7, have not been proved for very general classes of walks. The proofs here rely heavily on the fact that the increments of simple random walk are bounded and symmetric.

Diffusions, Markov Processes, and Martingales: Volume 1, Foundations

Author : L. C. G. Rogers
Publisher : Cambridge University Press
ISBN 13 : 1107717493
Total Pages : 412 pages
Book Rating : 4.1/5 (77 download)

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Book Synopsis Diffusions, Markov Processes, and Martingales: Volume 1, Foundations by : L. C. G. Rogers

Download or read book Diffusions, Markov Processes, and Martingales: Volume 1, Foundations written by L. C. G. Rogers and published by Cambridge University Press. This book was released on 2000-04-13 with total page 412 pages. Available in PDF, EPUB and Kindle. Book excerpt: Now available in paperback, this celebrated book has been prepared with readers' needs in mind, remaining a systematic guide to a large part of the modern theory of Probability, whilst retaining its vitality. The authors' aim is to present the subject of Brownian motion not as a dry part of mathematical analysis, but to convey its real meaning and fascination. The opening, heuristic chapter does just this, and it is followed by a comprehensive and self-contained account of the foundations of theory of stochastic processes. Chapter 3 is a lively and readable account of the theory of Markov processes. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.

Two-Dimensional Random Walk

Author : Serguei Popov
Publisher : Cambridge University Press
ISBN 13 : 1108472451
Total Pages : 224 pages
Book Rating : 4.1/5 (84 download)

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Book Synopsis Two-Dimensional Random Walk by : Serguei Popov

Download or read book Two-Dimensional Random Walk written by Serguei Popov and published by Cambridge University Press. This book was released on 2021-03-18 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: A visual, intuitive introduction in the form of a tour with side-quests, using direct probabilistic insight rather than technical tools.

Stochastic Calculus and Financial Applications

Author : J. Michael Steele
Publisher : Springer Science & Business Media
ISBN 13 : 1468493051
Total Pages : 303 pages
Book Rating : 4.4/5 (684 download)

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Book Synopsis Stochastic Calculus and Financial Applications by : J. Michael Steele

Download or read book Stochastic Calculus and Financial Applications written by J. Michael Steele and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 303 pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic calculus has important applications to mathematical finance. This book will appeal to practitioners and students who want an elementary introduction to these areas. From the reviews: "As the preface says, ‘This is a text with an attitude, and it is designed to reflect, wherever possible and appropriate, a prejudice for the concrete over the abstract’. This is also reflected in the style of writing which is unusually lively for a mathematics book." --ZENTRALBLATT MATH

Brownian Motion and Stochastic Calculus

Author : Ioannis Karatzas
Publisher : Springer
ISBN 13 : 1461209498
Total Pages : 490 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Brownian Motion and Stochastic Calculus by : Ioannis Karatzas

Download or read book Brownian Motion and Stochastic Calculus written by Ioannis Karatzas and published by Springer. This book was released on 2014-03-27 with total page 490 pages. Available in PDF, EPUB and Kindle. Book excerpt: A graduate-course text, written for readers familiar with measure-theoretic probability and discrete-time processes, wishing to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed, illustrated by results concerning representations of martingales and change of measure on Wiener space, which in turn permit a presentation of recent advances in financial economics. The book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The whole is backed by a large number of problems and exercises.

Intersections of Random Walks

Author : Gregoyr Lawler
Publisher : Birkhäuser
ISBN 13 : 9781461207726
Total Pages : 225 pages
Book Rating : 4.2/5 (77 download)

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Book Synopsis Intersections of Random Walks by : Gregoyr Lawler

Download or read book Intersections of Random Walks written by Gregoyr Lawler and published by Birkhäuser. This book was released on 2012-07-02 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: A more accurate title for this book would be "Problems dealing with the non-intersection of paths of random walks. " These include: harmonic measure, which can be considered as a problem of nonintersection of a random walk with a fixed set; the probability that the paths of independent random walks do not intersect; and self-avoiding walks, i. e. , random walks which have no self-intersections. The prerequisite is a standard measure theoretic course in probability including martingales and Brownian motion. The first chapter develops the facts about simple random walk that will be needed. The discussion is self-contained although some previous expo sure to random walks would be helpful. Many of the results are standard, and I have made borrowed from a number of sources, especially the ex cellent book of Spitzer [65]. For the sake of simplicity I have restricted the discussion to simple random walk. Of course, many of the results hold equally well for more general walks. For example, the local central limit theorem can be proved for any random walk whose increments have mean zero and finite variance. Some of the later results, especially in Section 1. 7, have not been proved for very general classes of walks. The proofs here rely heavily on the fact that the increments of simple random walk are bounded and symmetric.

Multiparameter Processes

Author : Davar Khoshnevisan
Publisher : Springer Science & Business Media
ISBN 13 : 0387216316
Total Pages : 590 pages
Book Rating : 4.3/5 (872 download)

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Book Synopsis Multiparameter Processes by : Davar Khoshnevisan

Download or read book Multiparameter Processes written by Davar Khoshnevisan and published by Springer Science & Business Media. This book was released on 2006-04-10 with total page 590 pages. Available in PDF, EPUB and Kindle. Book excerpt: Self-contained presentation: from elementary material to state-of-the-art research; Much of the theory in book-form for the first time; Connections are made between probability and other areas of mathematics, engineering and mathematical physics