Lévy Processes and Stochastic Calculus

Author: David Applebaum

Publisher: Cambridge University Press

ISBN: 9780521832632

Category: Mathematics

Page: 384

View: 1170

Graduate text decsribing two of the main tools for modern mathematical finance.

Lévy Processes and Stochastic Calculus

Author: David Applebaum

Publisher: Cambridge University Press

ISBN: 0521738652

Category: Mathematics

Page: 460

View: 8648

A fully revised and appended edition of this unique volume, which develops together these two important subjects.

Malliavin Calculus for Lévy Processes with Applications to Finance

Author: Giulia Di Nunno,Bernt Øksendal,Frank Proske

Publisher: Springer Science & Business Media

ISBN: 9783540785729

Category: Mathematics

Page: 418

View: 2979

This book is an introduction to Malliavin calculus as a generalization of the classical non-anticipating Ito calculus to an anticipating setting. It presents the development of the theory and its use in new fields of application.

Lévy Processes

Author: Jean Bertoin

Publisher: Cambridge University Press

ISBN: 9780521646321

Category: Mathematics

Page: 266

View: 9492

This 1996 book is a comprehensive account of the theory of Lévy processes; aimed at probability theorists.

Financial Modelling with Jump Processes

Author: Peter Tankov

Publisher: CRC Press

ISBN: 1135437947

Category: Mathematics

Page: 552

View: 9361

WINNER of a Riskbook.com Best of 2004 Book Award! During the last decade, financial models based on jump processes have acquired increasing popularity in risk management and option pricing. Much has been published on the subject, but the technical nature of most papers makes them difficult for nonspecialists to understand, and the mathematical tools required for applications can be intimidating. Potential users often get the impression that jump and Lévy processes are beyond their reach. Financial Modelling with Jump Processes shows that this is not so. It provides a self-contained overview of the theoretical, numerical, and empirical aspects involved in using jump processes in financial modelling, and it does so in terms within the grasp of nonspecialists. The introduction of new mathematical tools is motivated by their use in the modelling process, and precise mathematical statements of results are accompanied by intuitive explanations. Topics covered in this book include: jump-diffusion models, Lévy processes, stochastic calculus for jump processes, pricing and hedging in incomplete markets, implied volatility smiles, time-inhomogeneous jump processes and stochastic volatility models with jumps. The authors illustrate the mathematical concepts with many numerical and empirical examples and provide the details of numerical implementation of pricing and calibration algorithms. This book demonstrates that the concepts and tools necessary for understanding and implementing models with jumps can be more intuitive that those involved in the Black Scholes and diffusion models. If you have even a basic familiarity with quantitative methods in finance, Financial Modelling with Jump Processes will give you a valuable new set of tools for modelling market fluctuations.

Random Fragmentation and Coagulation Processes

Author: Jean Bertoin

Publisher: Cambridge University Press

ISBN: 1139459155

Category: Mathematics

Page: N.A

View: 8533

Fragmentation and coagulation are two natural phenomena that can be observed in many sciences and at a great variety of scales - from, for example, DNA fragmentation to formation of planets by accretion. This book, by the author of the acclaimed Lévy Processes, is the first comprehensive theoretical account of mathematical models for situations where either phenomenon occurs randomly and repeatedly as time passes. This self-contained treatment develops the models in a way that makes recent developments in the field accessible. Each chapter ends with a comments section in which important aspects not discussed in the main part of the text (often because the discussion would have been too technical and/or lengthy) are addressed and precise references are given. Written for readers with a solid background in probability, its careful exposition allows graduate students, as well as working mathematicians, to approach the material with confidence.

Probability and Information

An Integrated Approach

Author: David Applebaum

Publisher: Cambridge University Press

ISBN: 9780521555289

Category: Computers

Page: 212

View: 2129

This elementary introduction to probability theory and information theory provides a clear and systematic foundation to the subject; the author pays particular attention to the concept of probability via a highly simplified discussion of measures on Boolean algebras. He then applies the theoretical ideas to practical areas such as statistical inference, random walks, statistical mechanics, and communications modeling. Applebaum deals with topics including discrete and continuous random variables, entropy and mutual information, maximum entropy methods, the central limit theorem, and the coding and transmission of information. The author includes many examples and exercises that illustrate how the theory can be applied, e.g. to information technology. Solutions are available by email. This book is suitable as a textbook for beginning students in mathematics, statistics, or computer science who have some knowledge of basic calculus.

Brownian Motion and Stochastic Calculus

Author: Ioannis Karatzas,Steven Shreve

Publisher: Springer

ISBN: 1461209498

Category: Mathematics

Page: 470

View: 9595

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.

Probability and Stochastics

Author: Erhan Çınlar

Publisher: Springer Science & Business Media

ISBN: 9780387878591

Category: Mathematics

Page: 558

View: 9505

This text is an introduction to the modern theory and applications of probability and stochastics. The style and coverage is geared towards the theory of stochastic processes, but with some attention to the applications. In many instances the gist of the problem is introduced in practical, everyday language and then is made precise in mathematical form. The first four chapters are on probability theory: measure and integration, probability spaces, conditional expectations, and the classical limit theorems. There follows chapters on martingales, Poisson random measures, Levy Processes, Brownian motion, and Markov Processes. Special attention is paid to Poisson random measures and their roles in regulating the excursions of Brownian motion and the jumps of Levy and Markov processes. Each chapter has a large number of varied examples and exercises. The book is based on the author’s lecture notes in courses offered over the years at Princeton University. These courses attracted graduate students from engineering, economics, physics, computer sciences, and mathematics. Erhan Cinlar has received many awards for excellence in teaching, including the President’s Award for Distinguished Teaching at Princeton University. His research interests include theories of Markov processes, point processes, stochastic calculus, and stochastic flows. The book is full of insights and observations that only a lifetime researcher in probability can have, all told in a lucid yet precise style.

The Malliavin Calculus and Related Topics

Author: David Nualart

Publisher: Springer Science & Business Media

ISBN: 3540283293

Category: Mathematics

Page: 382

View: 2688

The Malliavin calculus is an infinite-dimensional differential calculus on a Gaussian space, developed to provide a probabilistic proof to Hörmander's sum of squares theorem but has found a range of applications in stochastic analysis. This book presents the features of Malliavin calculus and discusses its main applications. This second edition includes recent applications in finance and a chapter devoted to the stochastic calculus with respect to the fractional Brownian motion.

Introduction to Malliavin Calculus

Author: David Nualart,Eulalia Nualart

Publisher: Cambridge University Press

ISBN: 1107039126

Category: Business & Economics

Page: 246

View: 8819

This textbook offers a compact introductory course on Malliavin calculus, an active and powerful area of research. It covers recent applications, including density formulas, regularity of probability laws, central and non-central limit theorems for Gaussian functionals, convergence of densities and non-central limit theorems for the local time of Brownian motion. The book also includes a self-contained presentation of Brownian motion and stochastic calculus, as well as Lvy processes and stochastic calculus for jump processes. Accessible to non-experts, the book can be used by graduate students and researchers to develop their mastery of the core techniques necessary for further study.

Exotic Option Pricing and Advanced Lévy Models

Author: Andreas Kyprianou,Wim Schoutens,Paul Wilmott

Publisher: John Wiley & Sons

ISBN: 0470017201

Category: Business & Economics

Page: 344

View: 5075

Since around the turn of the millennium there has been a general acceptance that one of the more practical improvements one may make in the light of the shortfalls of the classical Black-Scholes model is to replace the underlying source of randomness, a Brownian motion, by a Lévy process. Working with Lévy processes allows one to capture desirable distributional characteristics in the stock returns. In addition, recent work on Lévy processes has led to the understanding of many probabilistic and analytical properties, which make the processes attractive as mathematical tools. At the same time, exotic derivatives are gaining increasing importance as financial instruments and are traded nowadays in large quantities in OTC markets. The current volume is a compendium of chapters, each of which consists of discursive review and recent research on the topic of exotic option pricing and advanced Lévy markets, written by leading scientists in this field. In recent years, Lévy processes have leapt to the fore as a tractable mechanism for modeling asset returns. Exotic option values are especially sensitive to an accurate portrayal of these dynamics. This comprehensive volume provides a valuable service for financial researchers everywhere by assembling key contributions from the world's leading researchers in the field. Peter Carr, Head of Quantitative Finance, Bloomberg LP. This book provides a front-row seat to the hottest new field in modern finance: options pricing in turbulent markets. The old models have failed, as many a professional investor can sadly attest. So many of the brightest minds in mathematical finance across the globe are now in search of new, more accurate models. Here, in one volume, is a comprehensive selection of this cutting-edge research. Richard L. Hudson, former Managing Editor of The Wall Street Journal Europe, and co-author with Benoit B. Mandelbrot of The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward

Stochastic Processes for Physicists

Understanding Noisy Systems

Author: Kurt Jacobs

Publisher: Cambridge University Press

ISBN: 1139486799

Category: Science

Page: 204

View: 928

Stochastic processes are an essential part of numerous branches of physics, as well as in biology, chemistry, and finance. This textbook provides a solid understanding of stochastic processes and stochastic calculus in physics, without the need for measure theory. In avoiding measure theory, this textbook gives readers the tools necessary to use stochastic methods in research with a minimum of mathematical background. Coverage of the more exotic Levy processes is included, as is a concise account of numerical methods for simulating stochastic systems driven by Gaussian noise. The book concludes with a non-technical introduction to the concepts and jargon of measure-theoretic probability theory. With over 70 exercises, this textbook is an easily accessible introduction to stochastic processes and their applications, as well as methods for numerical simulation, for graduate students and researchers in physics.

Hyperfinite Dirichlet Forms and Stochastic Processes

Author: Sergio Albeverio,Ruzong Fan,Frederik S. Herzberg

Publisher: Springer Science & Business Media

ISBN: 9783642196591

Category: Mathematics

Page: 284

View: 8063

This monograph treats the theory of Dirichlet forms from a comprehensive point of view, using "nonstandard analysis." Thus, it is close in spirit to the discrete classical formulation of Dirichlet space theory by Beurling and Deny (1958). The discrete infinitesimal setup makes it possible to study the diffusion and the jump part using essentially the same methods. This setting has the advantage of being independent of special topological properties of the state space and in this sense is a natural one, valid for both finite- and infinite-dimensional spaces. The present monograph provides a thorough treatment of the symmetric as well as the non-symmetric case, surveys the theory of hyperfinite Lévy processes, and summarizes in an epilogue the model-theoretic genericity of hyperfinite stochastic processes theory.

Markov Processes, Gaussian Processes, and Local Times

Author: Michael B. Marcus,Jay Rosen

Publisher: Cambridge University Press

ISBN: 1139458833

Category: Mathematics

Page: N.A

View: 6531

This book was first published in 2006. Written by two of the foremost researchers in the field, this book studies the local times of Markov processes by employing isomorphism theorems that relate them to certain associated Gaussian processes. It builds to this material through self-contained but harmonized 'mini-courses' on the relevant ingredients, which assume only knowledge of measure-theoretic probability. The streamlined selection of topics creates an easy entrance for students and experts in related fields. The book starts by developing the fundamentals of Markov process theory and then of Gaussian process theory, including sample path properties. It then proceeds to more advanced results, bringing the reader to the heart of contemporary research. It presents the remarkable isomorphism theorems of Dynkin and Eisenbaum and then shows how they can be applied to obtain new properties of Markov processes by using well-established techniques in Gaussian process theory. This original, readable book will appeal to both researchers and advanced graduate students.

Malliavin Calculus for Lévy Processes with Applications to Finance

Author: Giulia Di Nunno,Bernt Øksendal,Frank Proske

Publisher: Springer Science & Business Media

ISBN: 9783540785729

Category: Mathematics

Page: 418

View: 8230

This book is an introduction to Malliavin calculus as a generalization of the classical non-anticipating Ito calculus to an anticipating setting. It presents the development of the theory and its use in new fields of application.

Multidimensional Stochastic Processes as Rough Paths

Theory and Applications

Author: Peter K. Friz,Nicolas B. Victoir

Publisher: Cambridge University Press

ISBN: 1139487213

Category: Mathematics

Page: N.A

View: 521

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.

Stochastic Integration in Banach Spaces

Theory and Applications

Author: Vidyadhar Mandrekar,Barbara Rüdiger

Publisher: Springer

ISBN: 3319128531

Category: Mathematics

Page: 211

View: 950

Considering Poisson random measures as the driving sources for stochastic (partial) differential equations allows us to incorporate jumps and to model sudden, unexpected phenomena. By using such equations the present book introduces a new method for modeling the states of complex systems perturbed by random sources over time, such as interest rates in financial markets or temperature distributions in a specific region. It studies properties of the solutions of the stochastic equations, observing the long-term behavior and the sensitivity of the solutions to changes in the initial data. The authors consider an integration theory of measurable and adapted processes in appropriate Banach spaces as well as the non-Gaussian case, whereas most of the literature only focuses on predictable settings in Hilbert spaces. The book is intended for graduate students and researchers in stochastic (partial) differential equations, mathematical finance and non-linear filtering and assumes a knowledge of the required integration theory, existence and uniqueness results and stability theory. The results will be of particular interest to natural scientists and the finance community. Readers should ideally be familiar with stochastic processes and probability theory in general, as well as functional analysis and in particular the theory of operator semigroups. ​