1．PRELIMINARIES

1.1 Notations and Conventions

1.2 Measurability,Lp 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 μM

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 It？ Formula

5.3 Mutual Variation Process

5.4 Multi-dimensional It？ 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 Schr？dinger 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 It？ 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 It？ 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