It s Stochastic Calculus and Probability Theory

For example, a basic formula, called the Ito formula, is well known and widely used in fields as diverse as physics and economics. This volume contains 27 papers written by world-renowned probability theorists.

Author: Nobuyuki Ikeda

Publisher: Springer

ISBN: 4431685340

Category: Mathematics

Page: 422

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Professor Kiyosi Ito is well known as the creator of the modern theory of stochastic analysis. Although Ito first proposed his theory, now known as Ito's stochastic analysis or Ito's stochastic calculus, about fifty years ago, its value in both pure and applied mathematics is becoming greater and greater. For almost all modern theories at the forefront of probability and related fields, Ito's analysis is indispensable as an essential instrument, and it will remain so in the future. For example, a basic formula, called the Ito formula, is well known and widely used in fields as diverse as physics and economics. This volume contains 27 papers written by world-renowned probability theorists. Their subjects vary widely and they present new results and ideas in the fields where stochastic analysis plays an important role. Also included are several expository articles by well-known experts surveying recent developments. Not only mathematicians but also physicists, biologists, economists and researchers in other fields who are interested in the effectiveness of stochastic theory will find valuable suggestions for their research. In addition, students who are beginning their study and research in stochastic analysis and related fields will find instructive and useful guidance here. This volume is dedicated to Professor Ito on the occasion of his eightieth birthday as a token of deep appreciation for his great achievements and contributions. An introduction to and commentary on the scientific works of Professor Ito are also included.
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It s Stochastic Calculus and Probability Theory

For example, a basic formula, called the Ito formula, is well known and widely used in fields as diverse as physics and economics. This volume contains 27 papers written by world-renowned probability theorists.

Author: Nobuyuki Ikeda

Publisher: Springer Science & Business Media

ISBN: 9784431685326

Category: Mathematics

Page: 422

View: 167

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Professor Kiyosi Ito is well known as the creator of the modern theory of stochastic analysis. Although Ito first proposed his theory, now known as Ito's stochastic analysis or Ito's stochastic calculus, about fifty years ago, its value in both pure and applied mathematics is becoming greater and greater. For almost all modern theories at the forefront of probability and related fields, Ito's analysis is indispensable as an essential instrument, and it will remain so in the future. For example, a basic formula, called the Ito formula, is well known and widely used in fields as diverse as physics and economics. This volume contains 27 papers written by world-renowned probability theorists. Their subjects vary widely and they present new results and ideas in the fields where stochastic analysis plays an important role. Also included are several expository articles by well-known experts surveying recent developments. Not only mathematicians but also physicists, biologists, economists and researchers in other fields who are interested in the effectiveness of stochastic theory will find valuable suggestions for their research. In addition, students who are beginning their study and research in stochastic analysis and related fields will find instructive and useful guidance here. This volume is dedicated to Professor Ito on the occasion of his eightieth birthday as a token of deep appreciation for his great achievements and contributions. An introduction to and commentary on the scientific works of Professor Ito are also included.
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Stochastic Calculus and Financial Applications

Stochastic calculus has important applications to mathematical finance. This book will appeal to practitioners and students who want an elementary introduction to these areas.

Author: J. Michael Steele

Publisher: Springer Science & Business Media

ISBN: 9781468493054

Category: Mathematics

Page: 302

View: 642

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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
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Semimartingale Theory and Stochastic Calculus

The book emphasizes stochastic integration for semimartingales, characteristics of semimartingales, predictable representation properties and weak convergence of semimartingales.

Author: Sheng-Wu He

Publisher: Routledge

ISBN: 9781351416962

Category: Mathematics

Page: 400

View: 767

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Semimartingale Theory and Stochastic Calculus presents a systematic and detailed account of the general theory of stochastic processes, the semimartingale theory, and related stochastic calculus. The book emphasizes stochastic integration for semimartingales, characteristics of semimartingales, predictable representation properties and weak convergence of semimartingales. It also includes a concise treatment of absolute continuity and singularity, contiguity, and entire separation of measures by semimartingale approach. Two basic types of processes frequently encountered in applied probability and statistics are highlighted: processes with independent increments and marked point processes encountered frequently in applied probability and statistics. Semimartingale Theory and Stochastic Calculus is a self-contained and comprehensive book that will be valuable for research mathematicians, statisticians, engineers, and students.
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Elementary Stochastic Calculus with Finance in View

However, stochastic calculus is based on a deep mathematical theory. This book is suitable for the reader without a deep mathematical background.

Author: Thomas Mikosch

Publisher: World Scientific

ISBN: 9810235437

Category: Mathematics

Page: 212

View: 736

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Modelling with the Ito integral or stochastic differential equations has become increasingly important in various applied fields, including physics, biology, chemistry and finance. However, stochastic calculus is based on a deep mathematical theory. This book is suitable for the reader without a deep mathematical background. It gives an elementary introduction to that area of probability theory, without burdening the reader with a great deal of measure theory. Applications are taken from stochastic finance. In particular, the Black -- Scholes option pricing formula is derived. The book can serve as a text for a course on stochastic calculus for non-mathematicians or as elementary reading material for anyone who wants to learn about Ito calculus and/or stochastic finance.
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Stochastic Calculus

Whether your students are interested in probability, analysis, differential geometry or applications in operations research, physics, finance, or the many other areas to which the subject applies, you'll find that this text brings together ...

Author: Richard Durrett

Publisher: CRC Press

ISBN: 0849380715

Category: Mathematics

Page: 341

View: 120

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This compact yet thorough text zeros in on the parts of the theory that are particularly relevant to applications . It begins with a description of Brownian motion and the associated stochastic calculus, including their relationship to partial differential equations. It solves stochastic differential equations by a variety of methods and studies in detail the one-dimensional case. The book concludes with a treatment of semigroups and generators, applying the theory of Harris chains to diffusions, and presenting a quick course in weak convergence of Markov chains to diffusions. The presentation is unparalleled in its clarity and simplicity. Whether your students are interested in probability, analysis, differential geometry or applications in operations research, physics, finance, or the many other areas to which the subject applies, you'll find that this text brings together the material you need to effectively and efficiently impart the practical background they need.
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Stochastic Calculus

Whether your students are interested in probability, analysis, differential geometry or applications in operations research, physics, finance, or the many other areas to which the subject applies, you'll find that this text brings together ...

Author: Richard Durrett

Publisher: CRC Press

ISBN: 9781351413756

Category: Mathematics

Page: 341

View: 152

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This compact yet thorough text zeros in on the parts of the theory that are particularly relevant to applications . It begins with a description of Brownian motion and the associated stochastic calculus, including their relationship to partial differential equations. It solves stochastic differential equations by a variety of methods and studies in detail the one-dimensional case. The book concludes with a treatment of semigroups and generators, applying the theory of Harris chains to diffusions, and presenting a quick course in weak convergence of Markov chains to diffusions. The presentation is unparalleled in its clarity and simplicity. Whether your students are interested in probability, analysis, differential geometry or applications in operations research, physics, finance, or the many other areas to which the subject applies, you'll find that this text brings together the material you need to effectively and efficiently impart the practical background they need.
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An Introduction to Stochastic Differential Equations

This book will be very helpful to starting graduate students and strong undergraduates as well as to others who want to gain knowledge of stochastic differential equations. I recommend this book enthusiastically.

Author: Lawrence C. Evans

Publisher: American Mathematical Soc.

ISBN: 9781470410544

Category: Mathematics

Page: 151

View: 540

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These notes provide a concise introduction to stochastic differential equations and their application to the study of financial markets and as a basis for modeling diverse physical phenomena. They are accessible to non-specialists and make a valuable addition to the collection of texts on the topic. --Srinivasa Varadhan, New York University This is a handy and very useful text for studying stochastic differential equations. There is enough mathematical detail so that the reader can benefit from this introduction with only a basic background in mathematical analysis and probability. --George Papanicolaou, Stanford University This book covers the most important elementary facts regarding stochastic differential equations; it also describes some of the applications to partial differential equations, optimal stopping, and options pricing. The book's style is intuitive rather than formal, and emphasis is made on clarity. This book will be very helpful to starting graduate students and strong undergraduates as well as to others who want to gain knowledge of stochastic differential equations. I recommend this book enthusiastically. --Alexander Lipton, Mathematical Finance Executive, Bank of America Merrill Lynch This short book provides 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 concise and strongly focused upon the interplay between probabilistic intuition and mathematical rigor. Topics include a quick survey of measure theoretic probability theory, followed by an introduction to Brownian motion and the Ito stochastic calculus, and finally the theory of stochastic differential equations. The text also includes applications to partial differential equations, optimal stopping problems and options pricing. This book can be used as a text for senior undergraduates or beginning graduate students in mathematics, applied mathematics, physics, financial mathematics, etc., who want to learn the basics of stochastic differential equations. The reader is assumed to be fairly familiar with measure theoretic mathematical analysis, but is not assumed to have any particular knowledge of probability theory (which is rapidly developed in Chapter 2 of the book).
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Probability Theory III

This is a survey of stochastic calculus. The topics covered include: Brownian motion; the Ito integral; stochastic differential equations; Malliavin calculus; the general theory of random processes; and martingale theory.

Author: IUrii Vasil'evich Prokhorov

Publisher: Springer Science & Business Media

ISBN: 3540546871

Category: Business & Economics

Page: 253

View: 754

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This is a survey of stochastic calculus. The topics covered include: Brownian motion; the Ito integral; stochastic differential equations; Malliavin calculus; the general theory of random processes; and martingale theory.
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Probability Theory and Stochastic Processes with Applications Second Edition

This second edition has a unique approach that provides a broad and wide introduction into the fascinating area of probability theory.

Author: Oliver Knill

Publisher: World Scientific Publishing Company

ISBN: 9813109491

Category: Mathematics

Page: 500

View: 497

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This second edition has a unique approach that provides a broad and wide introduction into the fascinating area of probability theory. It starts on a fast track with the treatment of probability theory and stochastic processes by providing short proofs. The last chapter is unique as it features a wide range of applications in other fields like Vlasov dynamics of fluids, statistics of circular data, singular continuous random variables, Diophantine equations, percolation theory, random Schrödinger operators, spectral graph theory, integral geometry, computer vision, and processes with high risk.Many of these areas are under active investigation and this volume is highly suited for ambitious undergraduate students, graduate students and researchers.
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Brownian Motion and Stochastic Calculus

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.

Author: Ioannis Karatzas

Publisher: Springer

ISBN: 9781461209492

Category: Mathematics

Page: 470

View: 838

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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.
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Brownian Motion Martingales and Stochastic Calculus

This book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales.

Author: Jean-François Le Gall

Publisher: Springer

ISBN: 9783319310893

Category: Mathematics

Page: 273

View: 280

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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.
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Stochastic Processes and Filtering Theory

Taking the state-space approach to filtering, this text models dynamical systems by finite-dimensional Markov processes, outputs of stochastic difference, and differential equations.

Author: Andrew H. Jazwinski

Publisher: Courier Corporation

ISBN: 9780486318196

Category: Science

Page: 400

View: 624

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This unified treatment of linear and nonlinear filtering theory presents material previously available only in journals, and in terms accessible to engineering students. Its sole prerequisites are advanced calculus, the theory of ordinary differential equations, and matrix analysis. Although theory is emphasized, the text discusses numerous practical applications as well. Taking the state-space approach to filtering, this text models dynamical systems by finite-dimensional Markov processes, outputs of stochastic difference, and differential equations. Starting with background material on probability theory and stochastic processes, the author introduces and defines the problems of filtering, prediction, and smoothing. He presents the mathematical solutions to nonlinear filtering problems, and he specializes the nonlinear theory to linear problems. The final chapters deal with applications, addressing the development of approximate nonlinear filters, and presenting a critical analysis of their performance.
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Basic Stochastic Processes

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.

Author: Zdzislaw Brzezniak

Publisher: Springer Science & Business Media

ISBN: 9781447105336

Category: Mathematics

Page: 226

View: 178

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Stochastic 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.
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Stochastic Calculus

This book provides a comprehensive introduction to the theory of stochastic calculus and some of its applications.

Author: Paolo Baldi

Publisher: Springer

ISBN: 9783319622262

Category: Mathematics

Page: 627

View: 989

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This book provides a comprehensive introduction to the theory of stochastic calculus and some of its applications. It is the only textbook on the subject to include more than two hundred exercises with complete solutions. After explaining the basic elements of probability, the author introduces more advanced topics such as Brownian motion, martingales and Markov processes. The core of the book covers stochastic calculus, including stochastic differential equations, the relationship to partial differential equations, numerical methods and simulation, as well as applications of stochastic processes to finance. The final chapter provides detailed solutions to all exercises, in some cases presenting various solution techniques together with a discussion of advantages and drawbacks of the methods used. Stochastic Calculus will be particularly useful to advanced undergraduate and graduate students wishing to acquire a solid understanding of the subject through the theory and exercises. Including full mathematical statements and rigorous proofs, this book is completely self-contained and suitable for lecture courses as well as self-study.
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Introduction to Stochastic Integration

This introductory textbook provides a concise introduction to the Ito calculus. From the reviews: "Introduction to Stochastic Integration is exactly what the title says.

Author: Hui-Hsiung Kuo

Publisher: Springer Science & Business Media

ISBN: 9780387310572

Category: Mathematics

Page: 279

View: 404

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Also called Ito calculus, the theory of stochastic integration has applications in virtually every scientific area involving random functions. This introductory textbook provides a concise introduction to the Ito calculus. From the reviews: "Introduction to Stochastic Integration is exactly what the title says. I would maybe just add a ‘friendly’ introduction because of the clear presentation and flow of the contents." --THE MATHEMATICAL SCIENCES DIGITAL LIBRARY
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Introductory Stochastic Analysis for Finance and Insurance

While the text is self-contained, an introductory course in probability theory is beneficial to prospective readers. This book evolved from the author's experience as an instructor and has been thoroughly classroom-tested.

Author: X. Sheldon Lin

Publisher: John Wiley & Sons

ISBN: 9780471793205

Category: Mathematics

Page: 224

View: 459

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Incorporates the many tools needed for modeling and pricing infinance and insurance Introductory Stochastic Analysis for Finance and Insuranceintroduces readers to the topics needed to master and use basicstochastic analysis techniques for mathematical finance. The authorpresents the theories of stochastic processes and stochasticcalculus and provides the necessary tools for modeling and pricingin finance and insurance. Practical in focus, the book's emphasisis on application, intuition, and computation, rather thantheory. Consequently, the text is of interest to graduate students,researchers, and practitioners interested in these areas. While thetext is self-contained, an introductory course in probabilitytheory is beneficial to prospective readers. This book evolved from the author's experience as an instructor andhas been thoroughly classroom-tested. Following an introduction,the author sets forth the fundamental information and tools neededby researchers and practitioners working in the financial andinsurance industries: * Overview of Probability Theory * Discrete-Time stochastic processes * Continuous-time stochastic processes * Stochastic calculus: basic topics The final two chapters, Stochastic Calculus: Advanced Topics andApplications in Insurance, are devoted to more advanced topics.Readers learn the Feynman-Kac formula, the Girsanov's theorem, andcomplex barrier hitting times distributions. Finally, readersdiscover how stochastic analysis and principles are applied inpractice through two insurance examples: valuation of equity-linkedannuities under a stochastic interest rate environment andcalculation of reserves for universal life insurance. Throughout the text, figures and tables are used to help simplifycomplex theory and pro-cesses. An extensive bibliography opens upadditional avenues of research to specialized topics. Ideal for upper-level undergraduate and graduate students, thistext is recommended for one-semester courses in stochastic financeand calculus. It is also recommended as a study guide forprofessionals taking Causality Actuarial Society (CAS) and Societyof Actuaries (SOA) actuarial examinations.
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Problems and Solutions in Mathematical Finance

This volume introduces the reader to the basic stochastic calculus concepts required for the study of this important subject, providing a large number of worked examples which enable the reader to build the necessary foundation for more ...

Author: Eric Chin

Publisher: John Wiley & Sons

ISBN: 9781119965831

Category: Business & Economics

Page: 400

View: 222

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Brownian Motion Calculus

The inclusion of fully worked out exercises makes the book attractive for self study. Standard probability theory and ordinary calculus are the prerequisites. Summary slides for revision and teaching can be found on the book website.

Author: Ubbo F. Wiersema

Publisher: John Wiley & Sons

ISBN: 9780470021712

Category: Business & Economics

Page: 330

View: 913

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Brownian Motion Calculus presents the basics of Stochastic Calculus with a focus on the valuation of financial derivatives. It is intended as an accessible introduction to the technical literature. A clear distinction has been made between the mathematics that is convenient for a first introduction, and the more rigorous underpinnings which are best studied from the selected technical references. The inclusion of fully worked out exercises makes the book attractive for self study. Standard probability theory and ordinary calculus are the prerequisites. Summary slides for revision and teaching can be found on the book website.
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Multidimensional Stochastic Processes as Rough Paths

Examples, explanations and exercises make the book accessible to graduate students and researchers from a variety of fields.

Author: Peter K. Friz

Publisher: Cambridge University Press

ISBN: 9781139487214

Category: Mathematics

Page:

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Rough path analysis provides a fresh perspective on Ito's important theory of stochastic differential equations. Key theorems of modern stochastic analysis (existence and limit theorems for stochastic flows, Freidlin-Wentzell theory, the Stroock-Varadhan support description) can be obtained with dramatic simplifications. Classical approximation results and their limitations (Wong-Zakai, McShane's counterexample) receive 'obvious' rough path explanations. Evidence is building that rough paths will play an important role in the future analysis of stochastic partial differential equations and the authors include some first results in this direction. They also emphasize interactions with other parts of mathematics, including Caratheodory geometry, Dirichlet forms and Malliavin calculus. Based on successful courses at the graduate level, this up-to-date introduction presents the theory of rough paths and its applications to stochastic analysis. Examples, explanations and exercises make the book accessible to graduate students and researchers from a variety of fields.
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