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
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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.

Let Us Use White Noise

Author: Hida Takeyuki,Streit Ludwig
Publisher: World Scientific
ISBN: 9813220953
Category: Mathematics
Page: 232
View: 8206

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Why should we use white noise analysis? Well, one reason of course is that it fills that earlier gap in the tool kit. As Hida would put it, white noise provides us with a useful set of independent coordinates, parametrized by "time". And there is a feature which makes white noise analysis extremely user-friendly. Typically the physicist — and not only he — sits there with some heuristic ansatz, like e.g. the famous Feynman "integral", wondering whether and how this might make sense mathematically. In many cases the characterization theorem of white noise analysis provides the user with a sweet and easy answer. Feynman's "integral" can now be understood, the "It's all in the vacuum" ansatz of Haag and Coester is now making sense via Dirichlet forms, and so on in many fields of application. There is mathematical finance, there have been applications in biology, and engineering, many more than we could collect in the present volume. Finally, there is one extra benefit: when we internalize the structures of Gaussian white noise analysis we will be ready to meet another close relative. We will enjoy the important similarities and differences which we encounter in the Poisson case, championed in particular by Y Kondratiev and his group. Let us look forward to a companion volume on the uses of Poisson white noise. The present volume is more than a collection of autonomous contributions. The introductory chapter on white noise analysis was made available to the other authors early on for reference and to facilitate conceptual and notational coherence in their work.

Equations Involving Malliavin Calculus Operators

Applications and Numerical Approximation
Author: Tijana Levajković,Hermann Mena
Publisher: Springer
ISBN: 3319656783
Category: Mathematics
Page: 132
View: 3781

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This book provides a comprehensive and unified introduction to stochastic differential equations and related optimal control problems. The material is new and the presentation is reader-friendly. A major contribution of the book is the development of generalized Malliavin calculus in the framework of white noise analysis, based on chaos expansion representation of stochastic processes and its application for solving several classes of stochastic differential equations with singular data involving the main operators of Malliavin calculus. In addition, applications in optimal control and numerical approximations are discussed. The book is divided into four chapters. The first, entitled White Noise Analysis and Chaos Expansions, includes notation and provides the reader with the theoretical background needed to understand the subsequent chapters. In Chapter 2, Generalized Operators of Malliavin Calculus, the Malliavin derivative operator, the Skorokhod integral and the Ornstein-Uhlenbeck operator are introduced in terms of chaos expansions. The main properties of the operators, which are known in the literature for the square integrable processes, are proven using the chaos expansion approach and extended for generalized and test stochastic processes. Chapter 3, Equations involving Malliavin Calculus operators, is devoted to the study of several types of stochastic differential equations that involve the operators of Malliavin calculus, introduced in the previous chapter. Fractional versions of these operators are also discussed. Finally, in Chapter 4, Applications and Numerical Approximations are discussed. Specifically, we consider the stochastic linear quadratic optimal control problem with different forms of noise disturbances, operator differential algebraic equations arising in fluid dynamics, stationary equations and fractional versions of the equations studied – applications never covered in the extant literature. Moreover, numerical validations of the method are provided for specific problems."

Stochastic Integration in Banach Spaces

Theory and Applications
Author: Vidyadhar Mandrekar,Barbara Rüdiger
Publisher: Springer
ISBN: 3319128531
Category: Mathematics
Page: 211
View: 5361

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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. ​

PDE and Martingale Methods in Option Pricing

Author: Andrea Pascucci
Publisher: Springer Science & Business Media
ISBN: 9788847017818
Category: Mathematics
Page: 721
View: 6365

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This book offers an introduction to the mathematical, probabilistic and numerical methods used in the modern theory of option pricing. The text is designed for readers with a basic mathematical background. The first part contains a presentation of the arbitrage theory in discrete time. In the second part, the theories of stochastic calculus and parabolic PDEs are developed in detail and the classical arbitrage theory is analyzed in a Markovian setting by means of of PDEs techniques. After the martingale representation theorems and the Girsanov theory have been presented, arbitrage pricing is revisited in the martingale theory optics. General tools from PDE and martingale theories are also used in the analysis of volatility modeling. The book also contains an Introduction to Lévy processes and Malliavin calculus. The last part is devoted to the description of the numerical methods used in option pricing: Monte Carlo, binomial trees, finite differences and Fourier transform.

Levy Processes in Finance

Pricing Financial Derivatives
Author: Wim Schoutens
Publisher: Wiley
ISBN: 9780470851562
Category: Mathematics
Page: 200
View: 2906

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Financial mathematics has recently enjoyed considerable interest on account of its impact on the finance industry. In parallel, the theory of L?vy processes has also seen many exciting developments. These powerful modelling tools allow the user to model more complex phenomena, and are commonly applied to problems in finance. L?vy Processes in Finance: Pricing Financial Derivatives takes a practical approach to describing the theory of L?vy-based models, and features many examples of how they may be used to solve problems in finance. * Provides an introduction to the use of L?vy processes in finance. * Features many examples using real market data, with emphasis on the pricing of financial derivatives. * Covers a number of key topics, including option pricing, Monte Carlo simulations, stochastic volatility, exotic options and interest rate modelling. * Includes many figures to illustrate the theory and examples discussed. * Avoids unnecessary mathematical formalities. The book is primarily aimed at researchers and postgraduate students of mathematical finance, economics and finance. The range of examples ensures the book will make a valuable reference source for practitioners from the finance industry including risk managers and financial product developers.

A Course on Rough Paths

With an Introduction to Regularity Structures
Author: Peter K. Friz,Martin Hairer
Publisher: N.A
ISBN: 9783319083339
Page: 268
View: 1292

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Financial Modelling with Jump Processes

Author: Peter Tankov
Publisher: CRC Press
ISBN: 0203485211
Category: Mathematics
Page: 552
View: 1841

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WINNER of a 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.

Verallgemeinerte stochastische Prozesse

Modellierung und Anwendung technischer Rauschprozesse
Author: Stefan Schäffler
Publisher: Springer-Verlag
ISBN: 366254265X
Category: Mathematics
Page: 183
View: 4365

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Dieses Lehrbuch behandelt die in Natur- und Ingenieurwissenschaften eine zentrale Rolle spielenden Rauschprozesse, wie weißes Rauschen in der Raumsondenkommunikation oder thermisches Rauschen und Schrotrauschen in elektronischen Bauelementen.In dieser Form einzigartig, entwickelt der Autor die mathematische Theorie der verallgemeinerten stochastischen Prozesse und spricht dabei die Anwendung dieser mathematischen Objekte in der Praxis (z.B. Schaltkreissimulation, digitale Nachrichtenübertragung und Bildverarbeitung) an; somit dient dieses Lehrbuch auch als praxisrelevante Einführung in die Modellierung und Verwendung technischer Rauschprozesse. Die mathematische Modellierung von Rauschprozessen führt auf die Theorie stochastischer Prozesse auf Basis verallgemeinerter Funktionen (Distributionen), ohne die kein Handy funktionieren und Anwendungen wie die Simulation komplexer elektronischer Schaltungen unmöglich wäre.Für Anwender und interessierte Mathematiker bietet dieses Werk erstmals einen mathematisch fundierten Einblick in diese Thematik.

Advances in mathematics of finance

Author: Stefan Banach International Mathematical Center
Publisher: N.A
Category: Finance
Page: 249
View: 979

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"This volume contains 15 papers contributed by the participands of the 2nd General AMaMeF conference and Banach Center converence 'Advances in mathematics of finance' organized in Bȩdlewo, Poland from 30th April till 5th May, 2007. AMaMeF (Advances Mathematical Methods of Finance) is a scientific programme of the European Science Foundation for 2005-2010"--Preface (p. 5).


Deterministische Beobachtung und stochastische Filterung
Author: Karl Brammer,Gerhard Siffling
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3486785524
Category: Science
Page: 232
View: 1324

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Das Buch führt den Leser auf elementarem Wege in die Wahrscheinlichkeitsrechnung und in die Theorie der Zufallsprozesse ein, wobei keinerlei Vorkenntnisse auf diesem Gebiet vorausgesetzt werden. Schließlich wird gezeigt, wie sich die Eigenschaften eines Zufallsprozesses bei der Übertragung durch ein lineares System verändern und wie diese veränderten Eigenschaften berechnet werden können.

Quantenmechanik für Fortgeschrittene

Author: Franz Schwabl
Publisher: Springer-Verlag
ISBN: 3662096307
Category: Science
Page: 419
View: 9787

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Aufbauend auf der Quantenmechanik desselben Autors werden hier fortgeschrittene Themen behandelt: I Vielteilchensysteme, II Relativistische Wellengleichungen, III Relativistische Felder. Die in gewohnter Weise stringente mathematische Darstellung wird durch die Angabe aller Zwischenschritte, durch zahlreiche Anwendungsbeispiele im Text und Übungen ergänzt. Der Text legt insbesondere durch Darstellung der relativistischen Wellengleichungen und ihrer Symmetrieeigenschaften sowie der quantenfeldtheoretischen Grundlagen das Fundament für das weitere Studium von Festkörperphysik, Kern- und Elementarteilchenphysik.

Wahrscheinlichkeitstheorie und Stochastische Prozesse

Author: Michael Mürmann
Publisher: Springer-Verlag
ISBN: 364238160X
Category: Mathematics
Page: 428
View: 8462

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Dieses Lehrbuch beschäftigt sich mit den zentralen Gebieten einer maßtheoretisch orientierten Wahrscheinlichkeitstheorie im Umfang einer zweisemestrigen Vorlesung. Nach den Grundlagen werden Grenzwertsätze und schwache Konvergenz behandelt. Es folgt die Darstellung und Betrachtung der stochastischen Abhängigkeit durch die bedingte Erwartung, die mit der Radon-Nikodym-Ableitung realisiert wird. Sie wird angewandt auf die Theorie der stochastischen Prozesse, die nach der allgemeinen Konstruktion aus der Untersuchung von Martingalen und Markov-Prozessen besteht. Neu in einem Lehrbuch über allgemeine Wahrscheinlichkeitstheorie ist eine Einführung in die stochastische Analysis von Semimartingalen auf der Grundlage einer geeigneten Stetigkeitsbedingung mit Anwendungen auf die Theorie der Finanzmärkte. Das Buch enthält zahlreiche Übungen, teilweise mit Lösungen. Neben der Theorie vertiefen Anmerkungen, besonders zu mathematischen Modellen für Phänomene der Realität, das Verständnis.​

Stochastic Integrals

An Introduction
Author: Heinrich von Weizsäcker
Publisher: Springer-Verlag
ISBN: 3663139239
Category: Mathematics
Page: 332
View: 1593

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Author: Wolfgang Krull
Publisher: Springer-Verlag
ISBN: 3642870333
Category: Mathematics
Page: 160
View: 5225

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Dirichlet Forms and Analysis on Wiener Space

Author: Nicolas Bouleau,Francis Hirsch
Publisher: Walter de Gruyter
ISBN: 311085838X
Category: Mathematics
Page: 335
View: 5880

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The subject of this book is analysis on Wiener space by means of Dirichlet forms and Malliavin calculus. There are already several literature on this topic, but this book has some different viewpoints. First the authors review the theory of Dirichlet forms, but they observe only functional analytic, potential theoretical and algebraic properties. They do not mention the relation with Markov processes or stochastic calculus as discussed in usual books (e.g. Fukushima’s book). Even on analytic properties, instead of mentioning the Beuring-Deny formula, they discuss “carré du champ” operators introduced by Meyer and Bakry very carefully. Although they discuss when this “carré du champ” operator exists in general situation, the conditions they gave are rather hard to verify, and so they verify them in the case of Ornstein-Uhlenbeck operator in Wiener space later. (It should be noticed that one can easily show the existence of “carré du champ” operator in this case by using Shigekawa’s H-derivative.) In the part on Malliavin calculus, the authors mainly discuss the absolute continuity of the probability law of Wiener functionals. The Dirichlet form corresponds to the first derivative only, and so it is not easy to consider higher order derivatives in this framework. This is the reason why they discuss only the first step of Malliavin calculus. On the other hand, they succeeded to deal with some delicate problems (the absolute continuity of the probability law of the solution to stochastic differential equations with Lipschitz continuous coefficients, the domain of stochastic integrals (Itô-Ramer-Skorokhod integrals), etc.). This book focuses on the abstract structure of Dirichlet forms and Malliavin calculus rather than their applications. However, the authors give a lot of exercises and references and they may help the reader to study other topics which are not discussed in this book. Zentralblatt Math, Reviewer: S.Kusuoka (Hongo)


Einführung in die Wahrscheinlichkeitstheorie und Statistik
Author: Hans-Otto Georgii
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110386860
Category: Mathematics
Page: 448
View: 4012

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Due to the extremely positive reception of this textbook, it is now being published in its 5th edition. The book provides an introduction to the key ideas and elements of probability theory and statistics. Stochastic concepts, models, and methods are highlighted through typical application examples, then analyzed theoretically and systematically explored.