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Brownian Motion, Martingales, and Stochastic Calculus

Author: Jean-François Le Gall

Publisher: Springer

Published: 2016-04-28

Total Pages: 273

ISBN-13: 3319310895

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 and Stochastic Calculus

Author: Ioannis Karatzas

Publisher: Springer

Published: 2014-03-27

Total Pages: 490

ISBN-13: 1461209498

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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.

Continuous Martingales and Brownian Motion

Author: Daniel Revuz

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 608

ISBN-13: 3662064006

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"This is a magnificent book! Its purpose is to describe in considerable detail a variety of techniques used by probabilists in the investigation of problems concerning Brownian motion....This is THE book for a capable graduate student starting out on research in probability: the effect of working through it is as if the authors are sitting beside one, enthusiastically explaining the theory, presenting further developments as exercises." –BULLETIN OF THE L.M.S.

Stochastic Calculus

Author: Paolo Baldi

Publisher: Springer

Published: 2017-11-09

Total Pages: 627

ISBN-13: 3319622269

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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.

Stochastic Calculus for Fractional Brownian Motion and Applications

Author: Francesca Biagini

Publisher: Springer Science & Business Media

Published: 2008-02-17

Total Pages: 331

ISBN-13: 1846287979

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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.

Stochastic Calculus and Financial Applications

Author: J. Michael Steele

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 303

ISBN-13: 1468493051

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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

Random Walk, Brownian Motion, and Martingales

Author: Rabi Bhattacharya

Publisher: Springer Nature

Published: 2021-09-20

Total Pages: 396

ISBN-13: 303078939X

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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: René L. Schilling

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2014-06-18

Total Pages: 424

ISBN-13: 3110307308

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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 Calculus

Author: Ubbo F. Wiersema

Publisher: John Wiley & Sons

Published: 2008-12-08

Total Pages: 342

ISBN-13: 0470021705

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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

Author: Yuliya Mishura

Publisher: Springer

Published: 2008-04-12

Total Pages: 398

ISBN-13: 3540758739

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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.