Introduction 1 Preliminaries 1 Basic Definitions and Notation 2 Martingales 3 The Poisson Process and Brownian Motion 4 Levv Processes 5 Why the Usual Hypotheses? 6 Local Martingales 7 Stieltjes Integration and Change of Variables 8 Naive Stochastic Integration is Impossible Bibliographic Notes Exercises for Chapter 1 2 Semimartingales and Stochastic Integrals 1 Introduction to Semimartingales 2 Stability Properties of Semimartingales 3 Elementary Examples of Semimartingales 4 Stochastic Integrals 5 Properties of Stochastic Integrals 6 The Quadratic Variation of a Semimartingale 7 Ito's Formula (Change of Variables) 8 Applications of Ito's Formula Bibliographic Notes Exercises for Chapter 2 3 Semimartingales and Decomposable Processes 1 Introduction 2 The Classification of Stopping Times 3 The Doob-Meyer Decompositions 4 Quasimartingales 5 Compensators 6 The Fundamental Theorem of Local Martingales 7 Classical Semimartingales 8 Girsanov's Theorem 9 The Bichteler-Dellacherie Theorem Bibliographic Notes Exercises for Chapter 3 4 General Stochastic Integration and Local Times 1 Introduction 2 Stochastic Integration for Predictable Integrands 3 Martingale Representation 4 Martingale Duality and the Jacod-Yor Theorem on Martingale Representation 5 Examples of Martingale Representation 6 Stochastic Integration Depending on a Parameter 7 Local Times 8 Az6ma's Martingale 9 Sigma Martingales Bibliographic Notes Exercises for Chapter 4 5 Stochastic Differential Equations 1 Introduction 2 The H___p Norms for Semimartingales 3 Existence and Uniqueness of Solutions 4 Stability of Stochastic Differential Equations 5 Fisk-Stratonovich Integrals and Differential Equations 6 The Markov Nature of Solutions 7 Flows of Stochastic Differential Equations: Continuity and Differentiability 8 Flows as Diffeomorphisms: The Continuous Case 9 General Stochastic Exponentials and Linear Equations 10 Flows as Diffeomorphisms: The General Case 11 Eclectic Useful Results on Stochastic Differential Equations Bibliographic Notes Exercises for Chapter 5 6 Expansion of Filtrations 1 Introduction 2 Initial Expansions 3 Progressive Expansions 4 Time Reversal Bibliographic Notes Exercises for Chapter 6 References Subject Index
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