Introduction
Ⅰ 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 Ⅰ
Ⅱ 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 Ⅱ
Ⅲ 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 Ⅲ
Ⅳ 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 Ⅳ
Ⅴ Stochastic Differential Equations
1 Introduction
2 The Hp 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 Ⅴ
Ⅵ Expansion of Filtrations
1 Introduction
2 Initial Expansions
3 Progressive Expansions
4 Time Reversal
Bibliographic Notes
Exercises for Chapter Ⅵ
References
Subject Index
