隨機積分導論：第2版

PREFACE
PREFACE TO THE FIRST EDITION
ABBREVIATIONS AND SYMBOLS
1. PRELIMINARIES
1.1 Notations and Conventions
1.2 Measurability, L’’ Spaces and Monotone Class Theorems
1.3 Functions of Bounded Variation and Stieltjes Integrals
1.4 Probability Space, Random Variables, Filtration
1.5 Convergence, Conditioning
1.6 Stochastic Processes
1.7 Optional Times
1.8 Two Canonical Processes
1.9 Martingales
1.10 Local Martingales
1.11 Exercises

2. DEFINITION OF THE STOCHASTIC INTEGRAL
2.1 Introduction
2.2 Predictable Sets and Processes
2.3 Stochastic Intervals
2.4 Measure on the Predictable Sets
2.5 Definition of the Stochastic Integral
2.6 Extension to Local Integrators and Integrands
2.7 Substitution Formula
2.8 A Sufficient Condition for Extendability of λz
2.9 Exercises

3. EXTENSION OF THE PREDICTABLE INTEGRANDS
3.1 Introduction
3.2 Relationship between p, O, and Adapted Processes
3.3 Extension of the Integrands
3.4 A Historical Note
3.5 Exercises

4. QUADRATIC VARIATION PROCESS
4.1 Introduction
4.2 Definition and Characterization of Quadratic Variation
4.3 Properties of Quadratic Variation for an L2-martingale
4.4 Direct Definition of/JM
4.5 Decomposition of (M)2
4.6 A Limit Theorem
4.7 Exercises

5. THE ITO FORMULA
5.1 Introduction
5.2 One-dimensional Ito Formula
5.3 Mutual Variation Process
5.4 Multi-dimensional Ito Formula
5.5 Exercises

6. APPLICATIONS OF THE ITO FORMULA
6.1 Characterization of Brownian Motion
6.2 Exponential Processes
6.3 A Family of Martingales Generated by M
6.4 Feynman-Kac Functional and the Schrodinger Equation
6.5 Exercises

7. LOCAL TIME AND TANAKA’’S FORMULA
7.1 Introduction
7.2 Local Time
7.3 Tanaka’’s Formula
7.4 Proof of Lemma 7.2
7.5 Exercises

8. REFLECTED BROWNIAN MOTIONS
8.1 Introduction
8.2 Brownian Motion Reflected at Zero
8.3 Analytical Theory of Z via the It5 Formula
8.4 Approximations in Storage Theory
8.5 Reflected Brownian Motions in a Wedge
8.6 Alternative Derivation of Equation (8.7)
8.7 Exercises

9. GENERALIZED ITO FORMULA, CHANGE OF TIME AND MEASURE
9.1 Introduction
9.2 Generalized Ito Formula
9.3 Change of Time
9.4 Change of Measure
9.5 Exercises

10. STOCHASTIC DIFFERENTIAL EQUATIONS
10.1 Introduction
10.2 Existence and Uniqueness for Lipschitz Coefficients
10.3 Strong Markov Property of the Solution
10.4 Strong and Weak Solutions
10.5 Examples
10.6 Exercises
REFERENCES
INDEX