Martingales and Stochastic Integrals

This book provides an introduction to the rapidly expanding theory of stochastic integration and martingales.

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Author: P. E. Kopp

Publisher: Cambridge University Press

ISBN: 0521090334

Category: Mathematics

Page: 216

View: 408

This book provides an introduction to the rapidly expanding theory of stochastic integration and martingales. The treatment is close to that developed by the French school of probabilists, but is more elementary than other texts. The presentation is abstract, but largely self-contained and Dr Kopp makes fewer demands on the reader's background in probability theory than is usual. He gives a fairly full discussion of the measure theory and functional analysis needed for martingale theory, and describes the role of Brownian motion and the Poisson process as paradigm examples in the construction of abstract stochastic integrals. An appendix provides the reader with a glimpse of very recent developments in non-commutative integration theory which are of considerable importance in quantum mechanics. Thus equipped, the reader will have the necessary background to understand research in stochastic analysis. As a textbook, this account will be ideally suited to beginning graduate students in probability theory, and indeed it has evolved from such courses given at Hull University. It should also be of interest to pure mathematicians looking for a careful, yet concise introduction to martingale theory, and to physicists, engineers and economists who are finding that applications to their disciplines are becoming increasingly important.

Martingales and Stochastic Analysis

This book is a thorough and self-contained treatise of martingales as a tool in stochastic analysis, stochastic integrals and stochastic differential equations. The book is clearly written and details of proofs are worked out.

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Author: James Yeh

Publisher: World Scientific

ISBN: 981022477X

Category: Mathematics

Page: 501

View: 362

This book is a thorough and self-contained treatise of martingales as a tool in stochastic analysis, stochastic integrals and stochastic differential equations. The book is clearly written and details of proofs are worked out.

Nonlinear Filtering and Smoothing

Appropriate for upper-level undergraduates and graduate students, this volume addresses the fundamental concepts of martingales, stochastic integrals, and estimation. Written by an engineer for engineers, it emphasizes applications.

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Author: Venkatarama Krishnan

Publisher: Courier Corporation

ISBN: 9780486781839

Category: Science

Page: 336

View: 695

Appropriate for upper-level undergraduates and graduate students, this volume addresses the fundamental concepts of martingales, stochastic integrals, and estimation. Written by an engineer for engineers, it emphasizes applications.

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.

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Author: Jean-François Le Gall

Publisher: Springer

ISBN: 9783319310893

Category: Mathematics

Page: 273

View: 535

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.

Stochastic Integration and Differential Equations

The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises.

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Author: Philip Protter

Publisher: Springer

ISBN: 9783662100615

Category: Mathematics

Page: 415

View: 144

It has been 15 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it "a new approach". The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, the more general version of the Girsanov theorem due to Lenglart, the Kazamaki-Novikov criteria for exponential local martingales to be martingales, and a modern treatment of compensators. Chapter 4 treats sigma martingales (important in finance theory) and gives a more comprehensive treatment of martingale representation, including both the Jacod-Yor theory and Emery’s examples of martingales that actually have martingale representation (thus going beyond the standard cases of Brownian motion and the compensated Poisson process). New topics added include an introduction to the theory of the expansion of filtrations, a treatment of the Fefferman martingale inequality, and that the dual space of the martingale space H^1 can be identified with BMO martingales. Solutions to selected exercises are available at the web site of the author, with current URL http://www.orie.cornell.edu/~protter/books.html.

Introduction to Stochastic Calculus

This book sheds new light on stochastic calculus, the branch of mathematics that is most widely applied in financial engineering and mathematical finance.

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Author: Rajeeva L. Karandikar

Publisher: Springer

ISBN: 9789811083181

Category: Mathematics

Page: 441

View: 735

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

Introduction to Stochastic Integration

The contents of this monograph approximate the lectures I gave In a graduate course at Stanford University in the first half of 1981.

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Author: Chung

Publisher: Springer Science & Business Media

ISBN: 9781475791747

Category: Mathematics

Page: 192

View: 569

The contents of this monograph approximate the lectures I gave In a graduate course at Stanford University in the first half of 1981. But the material has been thoroughly reorganized and rewritten. The purpose is to present a modern version of the theory of stochastic in tegration, comprising but going beyond the classical theory, yet stopping short of the latest discontinuous (and to some distracting) ramifications. Roundly speaking, integration with respect to a local martingale with continuous paths is the primary object of study here. We have decided to include some results requiring only right continuity of paths, in order to illustrate the general methodology. But it is possible for the reader to skip these extensions without feeling lost in a wilderness of generalities. Basic probability theory inclusive of martingales is reviewed in Chapter 1. A suitably prepared reader should begin with Chapter 2 and consult Chapter 1 only when needed. Occasionally theorems are stated without proof but the treatmcnt is aimed at self-containment modulo the in evitable prerequisites. With considerable regret I have decided to omit a discussion of stochastic differential equations. Instead, some other ap plications of the stochastic calculus are given; in particular Brownian local time is treated in dctail to fill an unapparent gap in the literature. x I PREFACE The applications to storage theory discussed in Section 8. 4 are based on lectures given by J. Michael Harrison in my class.

Survey and Introduction to Modern Martingale Theory Stopping Times Semi Martingales and Stochastic Integration

This document is primarily about the work of the Strasbourg group led by Meyer and Delacherie and the development of the modern theory or semi-martingales and stochastic integration.

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Author:

Publisher:

ISBN: OCLC:227691516

Category:

Page: 192

View: 261

This document is primarily about the work of the Strasbourg group led by Meyer and Delacherie and the development of the modern theory or semi-martingales and stochastic integration. These developments have occurred over the last two decades and extend the theories of Doob and Ito. Unlike the sources, the first chapter introduces the subject in terms of a stochastic calculus for discrete parameter processes. In particular, discrete parameters point processes are defined and the discrete form of stochastic calculus is applied to them to obtain the nonlinear filtering formulas. There are two reasons for including the material in Chapter 1: First, discrete parameter point processes provide a useful tool for modeling a wide variety or dynamical systems; second, this material provides an elementary introduction to the remaining chapters. Chapter 2 introduces the notion of a filtered probability space stopping times relative to a filtration, stochastic intervals and defines previsible, optional and progressive stochastic processes. Stochastic point processes are considered in Chapter 3 along with Lebesgue-Stieltjes stochastic integrals, the simplest of the stochastic integrals. Chapter 4 contains one of the most basic concepts of modern martingale theory the dual previsible projection and local martingales are defined in Chapter 5. The construction of the stochastic integral of bounded previsible processes relative to semi-martingales is given in the final Chapter along with some applications to Brownian motion.

Stochastic Integrals

This text introduces at a moderate speed and in a thorough way the basic concepts of the theory of stochastic integrals and Ito calculus for sem i­ martingales.

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Author: Heinrich von Weizsäcker

Publisher: Vieweg+Teubner Verlag

ISBN: 3528063106

Category: Mathematics

Page: 332

View: 189

This text introduces at a moderate speed and in a thorough way the basic concepts of the theory of stochastic integrals and Ito calculus for sem i­ martingales. There are many reasons to study this subject. We are fascinated by the contrast between general measure theoretic arguments and concrete probabilistic problems, and by the own flavour of a new differential calculus. For the beginner, a lot of work is necessary to go through this text in detail. As areward it should enable her or hirn to study more advanced literature and to become at ease with a couple of seemingly frightening concepts. Already in this introduction, many enjoyable and useful facets of stochastic analysis show up. We start out having a glance at several elementary predecessors of the stochastic integral and sketching some ideas behind the abstract theory of semimartingale integration. Having introduced martingales and local martingales in chapters 2 - 4, the stochastic integral is defined for locally uniform limits of elementary processes in chapter S. This corresponds to the Riemann integral in one-dimensional analysis and it suffices for the study of Brownian motion and diffusion processes in the later chapters 9 and 12.

Mathematics of Financial Markets

This book explores the mathematics that underpins pricing models for derivative securities such as options, futures and swaps in modern markets.

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Author: Robert J Elliott

Publisher: Springer Science & Business Media

ISBN: 9781475771466

Category: Mathematics

Page: 292

View: 861

This book explores the mathematics that underpins pricing models for derivative securities such as options, futures and swaps in modern markets. Models built upon the famous Black-Scholes theory require sophisticated mathematical tools drawn from modern stochastic calculus. However, many of the underlying ideas can be explained more simply within a discrete-time framework. This is developed extensively in this substantially revised second edition to motivate the technically more demanding continuous-time theory.

Stochastic Integration Theory

This graduate level text covers the theory of stochastic integration, an important area of Mathematics that has a wide range of applications, including financial mathematics and signal processing.

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Author: Peter Medvegyev

Publisher: Oxford University Press on Demand

ISBN: 9780199215256

Category: Business & Economics

Page: 608

View: 566

This graduate level text covers the theory of stochastic integration, an important area of Mathematics that has a wide range of applications, including financial mathematics and signal processing. Aimed at graduate students in Mathematics, Statistics, Probability, Mathematical Finance, and Economics, the book not only covers the theory of the stochastic integral in great depth but also presents the associated theory (martingales, Levy processes) and important examples (Brownianmotion, Poisson process).

Stochastic Integrals

The text is suitable for economists, scientists, or researchers involved in probabilistic models and applied mathematics.

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Author: H. P. McKean

Publisher: Academic Press

ISBN: 9781483259239

Category: Mathematics

Page: 154

View: 739

Stochastic Integrals discusses one area of diffusion processes: the differential and integral calculus based upon the Brownian motion. The book reviews Gaussian families, construction of the Brownian motion, the simplest properties of the Brownian motion, Martingale inequality, and the law of the iterated logarithm. It also discusses the definition of the stochastic integral by Wiener and by Ito, the simplest properties of the stochastic integral according to Ito, and the solution of the simplest stochastic differential equation. The book explains diffusion, Lamperti's method, forward equation, Feller's test for the explosions, Cameron-Martin's formula, the Brownian local time, and the solution of dx=e(x) db + f(x) dt for coefficients with bounded slope. It also tackles Weyl's lemma, diffusions on a manifold, Hasminski's test for explosions, covering Brownian motions, Brownian motions on a Lie group, and Brownian motion of symmetric matrices. The book gives as example of a diffusion on a manifold with boundary the Brownian motion with oblique reflection on the closed unit disk of R squared. The text is suitable for economists, scientists, or researchers involved in probabilistic models and applied mathematics.

Stochastic Integrals for Nigh martingales

A stochastic integral is defined in which the integrand and the process with respect to which one integrates are stochastic, the process being supposed in a certain sense 'close' to being a martingale. (Author).

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Author: Laurence Chisholm Young

Publisher:

ISBN: OCLC:227583566

Category: Stochastic processes

Page: 23

View: 512

A stochastic integral is defined in which the integrand and the process with respect to which one integrates are stochastic, the process being supposed in a certain sense 'close' to being a martingale. (Author).