Stochastic integration and differential equations
Philip Protter - Collection Applications of mathematics
Résumé
It has been thirteen 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 even after thirteen years and many intervening texts, it seems worthwhile nevertheless to publish a second edition. We will no longer call it "a new approach" however.
The second edition has several significant changes. The most obvious is the addition of exercises for solution. These exercises are intended to supplement the text, and in no cases have lemmas needed in a proof been relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue University and Cornell University. Chapter three has been nearly 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 four treats sigma martingales which have become important in finance theory, as well as 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 H1 can be identified with BMO martingales. Last, there are of course small changes throughout the book.
Contents
- Introduction
- Preliminaries
- Basic Definitions and Notation
- Martingales
- The Poisson Process and Brownian Motion
- Lévy Processes
- Why the Usual Hypotheses?
- Local Martingales
- Stieltjes Integration and Change of Variables
- Naïve Stochastic Integration Is Impossible
- Bibliographic Notes
- Exercises for Chapter I
- Semimartingales and Stochastic Integrals
- Introduction to Semimartingales
- Stability Properties of Semimartingales
- Elementary Examples of Semimartingales
- Stochastic Integrals
- Properties of Stochastic Integrals
- The Quadratic Variation of a Semimartingale
- Itô's Formula (Change of Variables)
- Applications of Itô's Formula
- Bibliographic Notes
- Exercises for Chapter II
- Semimartingales and Decomposable Processes
- Introduction
- The Classification of Stopping Times
- The Doob-Meyer Decompositions
- Quasimartingales
- Compensators
- The Fundamental Theorem of Local Martingales
- Classical Semimartingales
- Girsanov's Theorem
- The Bichteler-Dellacherie Theorem
- Bibliographic Notes
- Exercises for Chapter III
- General Stochastic Integration and Local Times
- Introduction
- Stochastic Integration for Predictable Integrands
- Martingale Representation
- Martingale Duality and the Jacod-Yor Theorem on Martingale Representation
- Examples of Martingale Representation
- Stochastic Integration Depending on a Parameter
- Local Times
- Azéma's Martingale
- Sigma Martingales
- Bibliographic Notes
- Exercises for Chapter IV
- Stochastic Differential Equations
- Introduction
- The Hp Norms for Semimartingales
- Existence and Uniqueness of Solutions
- Stability of Stochastic Differential Equations
- Fisk-Stratonovich Integrals and Differential Equations
- The Markov Nature of Solutions
- Flows of Stochastic Differential Equations: Continuity and Differentiability
- Flows as Diffeomorphisms: The Continuous Case
- General Stochastic Exponentials and Linear Equations
- Flows as Diffeomorphisms: The General Case
- Eclectic Useful Results on Stochastic Differential Equations
- Bibliographic Notes
- Exercises for Chapter V
- Expansion of Filtrations
- Introduction
- Initial Expansions
- Progressive Expansions
- Time Reversal
- Bibliographic Notes
- Exercises for Chapter VI
- References
- Symbol Index
- Subject Index
L'auteur - Philip Protter
Cornell University, Ithaca, NY, USA
Caractéristiques techniques
| PAPIER | |
| Éditeur(s) | Springer |
| Auteur(s) | Philip Protter |
| Collection | Applications of mathematics |
| Parution | 16/12/2003 |
| Édition | 2eme édition |
| Nb. de pages | 428 |
| Format | 16 x 24 |
| Couverture | Relié |
| Poids | 735g |
| Intérieur | Noir et Blanc |
| EAN13 | 9783540003137 |
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