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Ⅰ Measure and Integration 1

1 Measurable Spaces 1

2 Measurable Functions 6

3 Measures 14

4 Integration 19

5 Transforms and Indefinite Integrals 29

6 Kernels and Product Spaces 37

Ⅱ Probability Spaces 49

1 Probability Spaces and Random Variables 50

2 Expectations 57

3 Lp-spaces and Uniform Integrability 70

4 Information and Determinability 75

5 Independence 82

Ⅲ Convergence 93

1 Convergence of Real Sequences 93

2 Almost Sure Convergence 97

3 Convergence in Probability 101

4 Convergence in Lp 105

5 Weak Convergence 109

6 Laws of Large Numbers 118

7 Convergence of Series 124

8 Central Limits 127

Ⅳ Conditioning 139

1 Conditional Expectations 139

2 Conditional Probabilities and Distributions 149

3 Conditional Independence 158

4 Construction of Probability Spaces 160

5 Special Constructions 166

Ⅴ Martingales and Stochastics 171

1 Filtrations and Stopping Times 171

2 Martingales 181

3 Martingale Transformations and Maxima 190

4 Martingale Convergence 199

5 Martingales in Continuous Time 213

6 Martingale Characterizations for Wiener and Poisson 225

7 Standard Filtrations and Modifications of Martingales 234

Ⅵ Poisson Random Measures 243

1 Random Measures 243

2 Poisson Random Measures 248

3 Transformations 263

4 Additive Random Measures and Lévy Processes 277

5 Poisson Processes 290

6 Poisson Integrals and Self-exciting Processes 298

Ⅶ Lévy Processes 313

1 Introduction 313

2 Stable Processes 329

3 Lévy Processes on Standard Settings 340

4 Characterizations for Wiener and Poisson 349

5 It？-Lévy Decomposition 354

6 Subordination 360

7 Increasing Lévy Processes 368

Ⅷ Brownian Motion 379

1 Introduction 379

2 Hitting Times and Recurrence Times 389

3 Hitting Times and Running Maximum 396

4 Wiener and its Maximum 399

5 Zeros,Local Times 408

6 Excursions 413

7 Path Properties 426

8 Existence 437

Ⅸ Markov Processes 443

1 Markov Property 444

2 It？ Diffusions 455

3 Jump-Diffusions 473

4 Markov Systems 498

5 Hunt Processes 505

6 Potentials and Excessive Functions 518

7 Appendix:Stochastic Integration 525

Notes and Comments 533

Bibliography 541

Index 551