1 Introduction 1
1.1 Stochastic Analogs of Classical Differential Equations 1
1.2 Filtering Problems 2
1.3 Stochastic Approach to Deterministic Boundary Value Problems 3
1.4 Optimal Stopping 3
1.5 Stochastic Control 4
1.6 Mathematical Finance 4
2 Some Mathematical Preliminaries 7
2.1 Probability Spaces，Random Variables and Stochastic Processes 7
2.2 An Important Example：Brownian Motion 12
Exercises 15
3 Ito Integrals 21
3.1 Construction of the Ito Integral 21
3.2 Some properties of the Ito integral 30
3.3 Extensions of the Ito integral 34
Exercises 37
4 The Ito Formula and the Martingale Representation Theorem 43
4.1 The 1-dimensional Ito formula 43
4.2 The Multi-dimensional Ito Formula 48
4.3 The Martingale Representation Theorem 49
Exercises 54
5 Stochastic Differential Equations 63
5.1 Examples and Some Solution Methods 63
5.2 An Existence and Uniqueness Result 68
5.3 Weak and Strong Solutions 72
Exercises 74
6 The Filtering Problem 83
6.1 Introduction 83
6.2 The 1-Dimensional Linear Filtering Problem 85
6.3 The Multidimensional Linear Filtering Problem 104
Exercises 105
7 Diffusions：Basic Properties 113
7.1 The Markov Property 113
7.2 The Strong Markov Property 116
7.3 The Generator of an Ito Diffusion 121
7.4 The Dynkin Formula 124
7.5 The Characteristic Operator 126
Exercises 128
8 Other Topics in Diffusion Theory 139
8.1 Kolmogorov’s Backward Equation.The Resolvent 139
8.2 The Feynman-Kac Formula.Killing 143
8.3 The Martingale Problem 146
8.4 When is an Ito Process a Diffusion？ 148
8.5 Random Time Change 153
8.6 The Girsanov Theorem 159
Exercises 168
9 Applications to Boundary Value Problems 175
9.1 The Combined Dirichlet-Poisson Problem.Uniqueness 175
9.2 The Dirichlet Problem.Regular Points 179
9.3 The Poisson Problem 190
Exercises 197
10 Application to Optimal Stopping 205
10.1 The Time-Homogeneous Case 205
10.2 The Time-Inhomogeneous Case 218
10.3 Optimal Stopping Problems Involving an Integral 222
10.4 Connection with Variational Inequalities 224
Exercises 228
11 Application to Stochastic Control 235
11.1 Statement of the Problem 235
11.2 The Hamilton-Jacobi-Bellman Equation 237
11.3 Stochastic control problems with terminal conditions 251
Exercises 252
12 Application to Mathematical Finance 261
12.1 Market，portfolio and arbitrage 261
12.2 Attainability and Completeness 271
12.3 Option Pricing 278
Exercises 298
Appendix A：Normal Random Variables 305
Appendix B：Conditional Expectation 309
Appendix C：Uniform Integrability and Martingale Convergence 311
Appendix D：An Approximation Result 315
Solutions and Additional Hints to Some of the Exercises 319
References 349
List of Frequently Used Notation and Symbols 357
Index 361