Stochastic differential equations: an introduction with applications Sixth Edition = 随机微分方程 第6版PDF电子书下载

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