1．PRELIMINARIES 1
1.1 Notations and Conventions 1
1.2 Measurability,Lp Spaces and Monotone Class Theorems 2
1.3 Functions of Bounded Variation and Stieltjes Integrals 4
1.4 Probability Space,Random Variables,Filtration 6
1.5 Convergence,Conditioning 7
1.6 Stochastic Processes 8
1.7 Optional Times 9
1.8 Two Canonical Processes 10
1.9 Martingales 13
1.10 Local Martingales 18
1.11 Exercises 21
2．DEFINITION OF THE STOCHASTIC INTEGRAL 23
2.1 Introduction 23
2.2 Predictable Sets and Processes 25
2.3 Stochastic Intervals 26
2.4 Measure on the Predictable Sets 32
2.5 Definition of the Stochastic Integral 34
2.6 Extension to Local Integrators and Integrands 43
2.7 Substitution Formula 48
2.8 A Sufficient Condition for Extendability of λz 50
2.9 Exercises 54
3．EXTENSION OF THE PREDICTABLE INTEGRANDS 57
3.1 Introduction 57
3.2 Relationship between P,O,and Adapted Processes 57
3.3 Extension of the Integrands 63
3.4 A Historical Note 71
3.5 Exercises 73
4．QUADRATIC VARIATION PROCESS 75
4.1 Introduction 75
4.2 Definition and Characterization of Quadratic Variation 75
4.3 Properties of Quadratic Variation for an L2-martingale 79
4.4 Direct Definition of μM 82
4.5 Decomposition of(M)2 86
4.6 A Limit Theorem 89
4.7 Exercises 90
5．THE ITO FORMULA 93
5.1 Introduction 93
5.2 One-dimensional It？ Formula 94
5.3 Mutual Variation Process 99
5.4 Multi-dimensional It？ Formula 109
5.5 Exercises 112
6．APPLICATIONS OF THE ITO FORMULA 117
6.1 Characterization of Brownian Motion 117
6.2 Exponential Processes 120
6.3 A Family of Martingales Generated by M 123
6.4 Feynman-Kac Functional and the Schr？dinger Equation 128
6.5 Exercises 136
7．LOCAL TIME AND TANAKA'S FORMULA 141
7.1 Introduction 141
7.2 Local Time 142
7.3 Tanaka's Formula 150
7.4 Proof of Lemma 7.2 153
7.5 Exercises 155
8．REFLECTED BROWNIAN MOTIONS 157
8.1 Introduction 157
8.2 Brownian Motion Reflected at Zero 158
8.3 Analytical Theory of Z via the It？ Formula 161
8.4 Approximations in Storage Theory 163
8.5 Reflected Brownian Motions in a Wedge 174
8.6 Alternative Derivation of Equation(8.7) 178
8.7 Exercises 181
9．GENERALIZED ITO FORMULA,CHANGE OF TIME AND MEASURE 183
9.1 Introduction 183
9.2 Generalized It？ Formula 184
9.3 Change of Time 187
9.4 Change of Measure 197
9.5 Exercises 214
10．STOCHASTIC DIFFERENTIAL EQUATIONS 217
10.1 Introduction 217
10.2 Existence and Uniqueness for Lipschitz Coefficients 220
10.3 Strong Markov Property of the Solution 235
10.4 Strong and Weak Solutions 243
10.5 Examples 252
10.6 Exercises 262
REFERENCES 265
INDEX 273