Brownian Motion And Stochastic Calculus

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Brownian Motion and Stochastic Calculus

Ioannis Karatzas,Steven Shreve
Brownian Motion and Stochastic Calculus Book PDF

Author : Ioannis Karatzas,Steven Shreve
Publisher : Springer
Page : 490 pages
File Size : 49,8 Mb
Release : 2014-03-27
Category : Mathematics
ISBN : 9781461209492

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Brownian Motion and Stochastic Calculus by Ioannis Karatzas,Steven Shreve Pdf

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.

Brownian Motion, Martingales, and Stochastic Calculus

Jean-François Le Gall
Brownian Motion Martingales and Stochastic Calculus Book PDF

Author : Jean-François Le Gall
Publisher : Springer
Page : 273 pages
File Size : 45,6 Mb
Release : 2016-04-28
Category : Mathematics
ISBN : 9783319310893

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Brownian Motion, Martingales, and Stochastic Calculus by Jean-François Le Gall Pdf

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.

Brownian Motion Calculus

Ubbo F. Wiersema
Brownian Motion Calculus Book PDF

Author : Ubbo F. Wiersema
Publisher : John Wiley & Sons
Page : 342 pages
File Size : 41,6 Mb
Release : 2008-12-08
Category : Business & Economics
ISBN : 9780470021705

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Brownian Motion Calculus by Ubbo F. Wiersema Pdf

BROWNIAN MOTION CALCULUS 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. The sequence of chapters starts with a description of Brownian motion, the random process which serves as the basic driver of the irregular behaviour of financial quantities. That exposition is based on the easily understood discrete random walk. Thereafter the gains from trading in a random environment are formulated in a discrete-time setting. The continuous-time equivalent requires a new concept, the Itō stochastic integral. Its construction is explained step by step, using the so-called norm of a random process (its magnitude), of which a motivated exposition is given in an Annex. The next topic is Itō’s formula for evaluating stochastic integrals; it is the random process counter part of the well known Taylor formula for functions in ordinary calculus. Many examples are given. These ingredients are then used to formulate some well established models for the evolution of stock prices and interest rates, so-called stochastic differential equations, together with their solution methods. Once all that is in place, two methodologies for option valuation are presented. One uses the concept of a change of probability and the Girsanov transformation, which is at the core of financial mathematics. As this technique is often perceived as a magic trick, particular care has been taken to make the explanation elementary and to show numerous applications. The final chapter discusses how computations can be made more convenient by a suitable choice of the so-called numeraire. 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 www.wiley.com/go/brownianmotioncalculus.

Stochastic Calculus for Fractional Brownian Motion and Related Processes

Yuliya Mishura
Stochastic Calculus for Fractional Brownian Motion and Related Processes Book PDF

Author : Yuliya Mishura
Publisher : Springer
Page : 398 pages
File Size : 42,6 Mb
Release : 2008-04-12
Category : Mathematics
ISBN : 9783540758730

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Stochastic Calculus for Fractional Brownian Motion and Related Processes by Yuliya Mishura Pdf

This volume examines the theory of fractional Brownian motion and other long-memory processes. Interesting topics for PhD students and specialists in probability theory, stochastic analysis and financial mathematics demonstrate the modern level of this field. It proves that the market with stock guided by the mixed model is arbitrage-free without any restriction on the dependence of the components and deduces different forms of the Black-Scholes equation for fractional market.

Stochastic Calculus for Fractional Brownian Motion and Applications

Francesca Biagini,Yaozhong Hu,Bernt Øksendal,Tusheng Zhang
Stochastic Calculus for Fractional Brownian Motion and Applications Book PDF

Author : Francesca Biagini,Yaozhong Hu,Bernt Øksendal,Tusheng Zhang
Publisher : Springer Science & Business Media
Page : 330 pages
File Size : 51,8 Mb
Release : 2008-02-17
Category : Mathematics
ISBN : 9781846287978

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Stochastic Calculus for Fractional Brownian Motion and Applications by Francesca Biagini,Yaozhong Hu,Bernt Øksendal,Tusheng Zhang Pdf

The purpose of this book is to present a comprehensive account of the different definitions of stochastic integration for fBm, and to give applications of the resulting theory. Particular emphasis is placed on studying the relations between the different approaches. Readers are assumed to be familiar with probability theory and stochastic analysis, although the mathematical techniques used in the book are thoroughly exposed and some of the necessary prerequisites, such as classical white noise theory and fractional calculus, are recalled in the appendices. This book will be a valuable reference for graduate students and researchers in mathematics, biology, meteorology, physics, engineering and finance.

Brownian Motion

René L. Schilling,Lothar Partzsch
Brownian Motion Book PDF

Author : René L. Schilling,Lothar Partzsch
Publisher : Walter de Gruyter GmbH & Co KG
Page : 424 pages
File Size : 54,8 Mb
Release : 2014-06-18
Category : Mathematics
ISBN : 9783110307306

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Brownian Motion by René L. Schilling,Lothar Partzsch Pdf

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.

Brownian Motion

René L. Schilling
Brownian Motion Book PDF

Author : René L. Schilling
Publisher : Walter de Gruyter GmbH & Co KG
Page : 533 pages
File Size : 42,6 Mb
Release : 2021-09-07
Category : Mathematics
ISBN : 9783110741278

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Brownian Motion by René L. Schilling Pdf

Stochastic processes occur everywhere in the sciences, economics and engineering, and they need to be understood by (applied) mathematicians, engineers and scientists alike. This book gives a gentle introduction to Brownian motion and stochastic processes, in general. Brownian motion plays a special role, since it shaped the whole subject, displays most random phenomena while being still easy to treat, and is used in many real-life models. Im this new edition, much material is added, and there are new chapters on ''Wiener Chaos and Iterated Itô Integrals'' and ''Brownian Local Times''.

Introduction to Stochastic Calculus with Applications

Fima C. Klebaner
Introduction to Stochastic Calculus with Applications Book PDF

Author : Fima C. Klebaner
Publisher : Imperial College Press
Page : 431 pages
File Size : 51,8 Mb
Release : 2005
Category : Mathematics
ISBN : 9781860945557

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Introduction to Stochastic Calculus with Applications by Fima C. Klebaner Pdf

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.

Stochastic Calculus

Richard Durrett
Stochastic Calculus Book PDF

Author : Richard Durrett
Publisher : CRC Press
Page : 356 pages
File Size : 43,7 Mb
Release : 2018-03-29
Category : Mathematics
ISBN : 9781351413749

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Stochastic Calculus by Richard Durrett Pdf

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.

Stochastic Calculus and Financial Applications

J. Michael Steele
Stochastic Calculus and Financial Applications Book PDF

Author : J. Michael Steele
Publisher : Springer Science & Business Media
Page : 303 pages
File Size : 53,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781468493054

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Stochastic Calculus and Financial Applications by J. Michael Steele Pdf

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

Stochastic Calculus

Paolo Baldi
Stochastic Calculus Book PDF

Author : Paolo Baldi
Publisher : Springer
Page : 627 pages
File Size : 42,5 Mb
Release : 2017-11-09
Category : Mathematics
ISBN : 9783319622262

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Stochastic Calculus by Paolo Baldi Pdf

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.

Stochastic Calculus

Mircea Grigoriu
Stochastic Calculus Book PDF

Author : Mircea Grigoriu
Publisher : Springer Science & Business Media
Page : 784 pages
File Size : 40,6 Mb
Release : 2013-12-11
Category : Mathematics
ISBN : 9780817682286

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Stochastic Calculus by Mircea Grigoriu Pdf

Algebraic, differential, and integral equations are used in the applied sciences, en gineering, economics, and the social sciences to characterize the current state of a physical, economic, or social system and forecast its evolution in time. Generally, the coefficients of and/or the input to these equations are not precisely known be cause of insufficient information, limited understanding of some underlying phe nomena, and inherent randonmess. For example, the orientation of the atomic lattice in the grains of a polycrystal varies randomly from grain to grain, the spa tial distribution of a phase of a composite material is not known precisely for a particular specimen, bone properties needed to develop reliable artificial joints vary significantly with individual and age, forces acting on a plane from takeoff to landing depend in a complex manner on the environmental conditions and flight pattern, and stock prices and their evolution in time depend on a large number of factors that cannot be described by deterministic models. Problems that can be defined by algebraic, differential, and integral equations with random coefficients and/or input are referred to as stochastic problems. The main objective of this book is the solution of stochastic problems, that is, the determination of the probability law, moments, and/or other probabilistic properties of the state of a physical, economic, or social system. It is assumed that the operators and inputs defining a stochastic problem are specified.

Stochastic Calculus for Finance I

Steven Shreve
Stochastic Calculus for Finance I Book PDF

Author : Steven Shreve
Publisher : Springer Science & Business Media
Page : 212 pages
File Size : 54,6 Mb
Release : 2005-06-28
Category : Mathematics
ISBN : 0387249680

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Stochastic Calculus for Finance I by Steven Shreve Pdf

Developed for the professional Master's program in Computational Finance at Carnegie Mellon, the leading financial engineering program in the U.S. Has been tested in the classroom and revised over a period of several years Exercises conclude every chapter; some of these extend the theory while others are drawn from practical problems in quantitative finance

Stochastic Integrals

Henry P. McKean
Stochastic Integrals Book PDF

Author : Henry P. McKean
Publisher : American Mathematical Soc.
Page : 159 pages
File Size : 46,7 Mb
Release : 2005
Category : Brownian movements
ISBN : 9780821838877

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Stochastic Integrals by Henry P. McKean Pdf

The AMS is excited to bring this volume, originally published in 1969, back into print. This well-written book has been used for many years to learn about stochastic integrals. The author starts with the presentation of Brownian motion, then deals with stochastic integrals and differentials, including the famous Ito lemma. The rest of the book is devoted to various topics of stochastic integral equations and stochastic integral equations on smooth manifolds. E. B. Dynkin wrote aboutthe original edition in Mathematical Reviews: "This little book is a brilliant introduction to an important boundary field between the theory of probability and that of differential equations ... differential and integral calculus based upon Brownian motion." These words continue to ring true today.This classic book is ideal for supplementary reading or independent study. It is suitable for graduate students and researchers interested in probability, stochastic processes, and their applications.

Nonlinear Expectations and Stochastic Calculus under Uncertainty

Shige Peng
Nonlinear Expectations and Stochastic Calculus under Uncertainty Book PDF

Author : Shige Peng
Publisher : Springer Nature
Page : 212 pages
File Size : 48,6 Mb
Release : 2019-09-09
Category : Mathematics
ISBN : 9783662599037

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Nonlinear Expectations and Stochastic Calculus under Uncertainty by Shige Peng Pdf

This book is focused on the recent developments on problems of probability model uncertainty by using the notion of nonlinear expectations and, in particular, sublinear expectations. It provides a gentle coverage of the theory of nonlinear expectations and related stochastic analysis. Many notions and results, for example, G-normal distribution, G-Brownian motion, G-Martingale representation theorem, and related stochastic calculus are first introduced or obtained by the author. This book is based on Shige Peng’s lecture notes for a series of lectures given at summer schools and universities worldwide. It starts with basic definitions of nonlinear expectations and their relation to coherent measures of risk, law of large numbers and central limit theorems under nonlinear expectations, and develops into stochastic integral and stochastic calculus under G-expectations. It ends with recent research topic on G-Martingale representation theorem and G-stochastic integral for locally integrable processes. With exercises to practice at the end of each chapter, this book can be used as a graduate textbook for students in probability theory and mathematical finance. Each chapter also concludes with a section Notes and Comments, which gives history and further references on the material covered in that chapter. Researchers and graduate students interested in probability theory and mathematical finance will find this book very useful.