随机过程用的极限定理 第2版 英文PDF电子书下载

Chapter Ⅰ．The General Theory of Stochastic Processes,Semimartingales and Stochastic Integrals 1

1．Stochastic Basis,Stopping Times,Optionalσ-Field,Martingales 1

1a．Stochastic Basis 2

1b．Stopping Times 4

1c．The Optional σ-Field 5

1d．The Localization Procedure 8

1e．Martingales 10

1f．The Discrete Case 13

2．Predictable σ-Field,Predictable Times 16

2a．The Predictable σ-Field 16

2b．Predictable Times 17

2c．Totally Inaccessible Stopping Times 20

2d．Predictable Projection 22

2e．The Discrete Case 25

3．Increasing Processes 27

3a．Basic Properties 27

3b．Doob-Meyer Decomposition and Compensators of Increasing Processes 32

3c．Lenglart Domination Property 35

3d．The Discrete Case 36

4．Semimartingales and Stochastic Integrals 38

4a．Locally Square-Integrable Martingales 38

4b．Decompositions of a Local Martingale 40

4c．Semimartingales 43

4d．Construction of the Stochastic Integral 46

4e．Quadratic Variation of a Semimartingale and Ito's Formula 51

4f．Doléans-Dade Exponential Formula 58

4g．The Discrete Case 62

Chapter Ⅱ．Characteristics of Semimartingales and Processes with Independent Increments 64

1．Random Measures 64

1a．General Random Measures 65

1b．Integer-Valued Random Measures 68

1c．A Fundamental Example：Poisson Measures 70

1d．Stochastic Integral with Respect to a Random Measure 71

2．Characteristics of Semimartingales 75

2a．Definition of the Characteristics 75

2b．Integrability and Characteristics 81

2c．A Canonical Representation for Semimartingales 84

2d．Characteristics and Exponential Formula 85

3．Some Examples 91

3a．The Discrete Case 91

3b．More on the Discrete Case 93

3c．The"One-Point"Point Process and Empirical Processes 97

4．Semimartingales with Independent Increments 101

4a．Wiener Processes 102

4b．Poisson Processes and Poisson Random Measures 103

4c．Processes with Independent Increments and Semimartingales 106

4d．Gaussian Martingales 111

5．Processes with Independent Increments Which Are Not Semimartingales 114

5a．The Results 114

5b．The Proofs 116

6．Processes with Conditionally Independent Increments 124

7．Progressive Conditional Continuous PIIs 128

8．Semimartingales,Stochastic Exponential and Stochastic Logarithm 134

8a．More About Stochastic Exponential and Stochastic Logarithm 134

8b．Multiplicative Decompositions and Exponentially Special Semimartingales 138

Chapter Ⅲ．Martingale Problems and Changes of Measures 142

1．Martingale Problems and Point Processes 143

1a．General Martingale Problems 143

1b．Martingale Problems and Random Measures 144

1c．Point Processes and Multivariate Point Processes 146

2．Martingale Problems and Semimartingales 151

2a．Formulation of the Problem 152

2b．Example：Processes with Independent Increments 154

2c．Diflusion Processes and Diffusion Processes with Jumps 155

2d．Local Uniqueness 159

3．Absolutely Continuous Changes of Measures 165

3a．The Density Process 165

3b．Girsanov's Theorem for Local Martingales 168

3c．Girsanoy's Theorem for Random Measures 170

3d．Girsanov's Theorem for Semimartingales 172

3e．The Discrete Case 177

4．Representation Theorem for Martingales 179

4a．Stochastic Integrals with Respect to a Multi-Dimensional Continuous Local Martingale 179

4b．Projection of a Local Martingale on a Random Measure 182

4c．The Representation Property 185

4d．The Fundamental Representation Theorem 187

5．Absolutely Continuous Change of Measures：Explicit Computation of the Density Process 191

5a．All P-Martingales Have the Representation Property Relative to X 192

5b．P′Has the Local Uniqueness Property 196

5c．Examples 200

6．Integrals of Vector-Valued Processes and σ-martingales 203

6a．Stochastic Integrals with Respect to a Multi-Dimensional Locally Square-integrable Martingale 204

6b．Integrals with Respect to a Multi-Dimensional Process of Locally Finite Variation 206

6c．Stochastic Integrals with Respect to a Multi-Dimensional Semimartingale 207

6d．Stochastic Integrals：A Predictable Criterion 212

6e．∑-localization and σ-martingales 214

7．Laplace Cumulant Processes and Esscher's Change of Measures 219

7a．Laplace Cumulant Processes of Exponentially Special Semimartingales 219

7b．Esscher Change of Measure 222

Chapter Ⅳ．Hellinger Processes,Absolute Continuity and Singularity of Measures 227

1．Hellinger Integrals and Hellinger Processes 228

1a．Kakutani-Hellinger Distance and Hellinger Integrals 228

1b．Hellinger Processes 230

1c．Computation of Hellinger Processes in Terms of the Density Processes 234

1d．Some Other Processes of Interest 237

1e．The Discrete Case 242

2．Predictable Criteria for Absolute Continuity and Singularity 245

2a．Statement of the Results 245

2b．The Proofs 248

2c．The Discrete Case 252

3．Hellinger Processes for Solutions of Martingale Problems 254

3a．The General Setting 255

3b．The Case Where P and P′Are Dominated by a Measure Having the Martingale Representation Property 257

3c．The Case Where Local Uniqueness Holds 266

4．Examples 272

4a．Point Processes and Multivariate Point Processes 272

4b．Generalized Diffusion Processes 275

4c．Processes with Independent Increments 277

Chapter Ⅴ．Contiguity,Entire Separation,Convergence in Variation 284

1．Contiguity and Entire Separation 284

1a．General Facts 284

1b．Contiguity and Filtrations 290

2．Predictable Criteria for Contiguity and Entire Separation 291

2a．Statements of the Results 291

2b．The Proofs 294

2c．The Discrete Case 301

3．Examples 304

3a．Point Processes 304

3b．Generalized Diffusion Processes 305

3c．Processes with Independent Increments 306

4．Variation Metric 309

4a．Variation Metric and Hellinger Integrals 310

4b．Variation Metric and Hellinger Processes 312

4c．Examples：Point Processes and Multivariate Point Processes 318

4d．Example：Generalized Diffusion Processes 322

Chapter Ⅵ．Skorokhod Topology and Convergence of Processes 324

1．The Skorokhod Topology 325

1a．Introduction and Notation 325

1b．The Skorokhod Topology：Definition and Main Results 327

1c．Proof of Theorem 1.14 329

2．Continuity for the Skorokhod Topology 337

2a．Continuity Properties of some Functions 337

2b．Increasing Functions and the Skorokhod Topology 342

3．Weak Convergence 347

3a．Weak Convergence of Probability Measures 347

3b．Application to Càdlàg Processes 348

4．Criteria for Tightness：The Quasi-Left Continuous Case 355

4a．Aldous'Criterion for Tightness 356

4b．Application to Martingales and Semimartingales 358

5．Criteria for Tightness：The General Case 362

5a．Criteria for Semimartingales 362

5b．An Auxiliary Result 365

5c．Proof of Theorem 5.17 367

6．Convergence,Quadratic Variation,Stochastic Integrals 376

6a．The P-UT Condition 377

6b．Tightness and the P-UT Property 382

6c．Convergence of Stochastic Integrals and Quadratic Variation 382

6d．Some Additional Results 386

Chapter Ⅶ．Convergence of Processes with Independent Increments 389

1．Introduction to Functional Limit Theorems 390

2．Finite-Dimensional Convergence 394

2a．Convergence of Infinitely Divisible Distributions 394

2b．Some Lemmas on Characteristic Functions 398

2c．Convergence of Rowwise Independent Triangular Arrays 402

2d．Finite-Dimensional Convergence of PII-Semimartingales to a PII Without Fixed Time of Discontinuity 408

3．Functional Convergence and Characteristics 413

3a．The Results 414

3b．Sufficient Condition for Convergence Under2.48 418

3c．Necessary Condition for Convergence 418

3d．Sufficient Condition for Convergence 424

4．More on the General Case 428

4a．Convergence ofNon-Infinitesimal Rowwise Independent　Arrays 428

4b．Finite-Dimensional Convergence for General PII 436

4c．Another Necessary and Sufficient Condition for Functional Convergence 439

5．The Central Limit Theorem 444

5a．The Lindeberg-Feller Theorem 445

5b．Zolotarev's Type Theorems 446

5c．Finite-Dimensional Convergence of PII's to a Gaussian Martingale 450

5d．Functional Convergence of PII's to a Gaussian Martingale 452

Chapter Ⅷ．Convergence to a Process with Independent Increments 456

1．Finite-Dimensional Convergence,a General Theorem 456

1a．Description of the Setting for This Chapter 456

1b．The Basic Theorem 457

1c．Remarks and Comments 459

2．Convergence to a PII Without Fixed Time of Discontinuity 460

2a．Finite-Dimensional Convergence 461

2b．Functional Convergence 464

2c．Application to Triangular Arrays 465

2d．Other Conditions for Convergence 467

3．Applications 469

3a．Central Limit Theorem：Necessary and Sufficient Conditions 470

3b．Central Limit Theorem：The Martingale Case 473

3c．Central Limit Theorem for Triangular Arrays 477

3d．Convergence of Point Processes 478

3e．Normed Sums of I.I.D.Semimartingales 481

3f．Limit Theorems for Functionals of Markov Processes 486

3g．Limit Theorems for Stationary Processes 489

4．Convergence to a General Process with Independent Increments 499

4a．Proof of Theorem 4.1 When the Characteristic Function of Xt Vanishes Almost Nowhere 501

4b．Convergence of Point Processes 503

4c．Convergence to a Gaussian Martingale 504

5．Convergence to a Mixture of PII's,Stable Convergence and Mixing Convergence 506

5a．Convergence to a Mixture of PII's 506

5b．More on the Convergence to a Mixture of PII's 510

5c．Stable Convergence 512

5d．Mixing Convergence 518

5e．Application to Stationary Processes 519

Chapter Ⅸ．Convergence to a Semimartingale 521

1．Limits of Martingales 521

1a．The Bounded Case 522

1b．The Unbounded Case 524

2．Identification of the Limit 527

2a．Introductory Remarks 527

2b．Identification of the Limit：The Main Result 530

2c．Identification of the Limit Via Convergence of the Characteristics 533

2d．Application：Existence of Solutions to Some Martingale Problems 535

3．Limit Theorems for Semimartingales 540

3a．Tightness of the Sequence(Xn) 541

3b．Limit Theorems：The Bounded Case 546

3c．Limit Theorems：The Locally Bounded Case 550

4．Applications 554

4a．Convergence of Diffusion Processes with Jumps 554

4b．Convergence of Step Markov Processes to Diffusions 557

4c．Empirical Distributions and Brownian Bridge 560

4d．Convergence to a Continuous Semimartingale：Necessary and Sufficient Conditions 561

5．Convergence of Stochastic Integrals 564

5a．Characteristics of Stochastic Integrals 564

5b．Statement of the Results 567

5c．The Proofs 570

6．Stability for Stochastic Differential Equation 575

6a．Auxiliary Results 576

6b．Stochastic Differential Equations 577

6c．Stability 578

7．Stable Convergence to a Progressive Conditional Continuous PII 583

7a．A General Result 583

7b．Convergence of Discretized Processes 589

Chapter Ⅹ．Limit Theorems,Density Processes and Contiguity 592

1．Convergence of the Density Processes to a Continuous Process 593

1a．Introduction,Statement of the Main Results 593

1b．An Auxiliary Computation 597

1c．Proofs of Theorems 1.12 and 1.16 603

1d．Convergence to the Exponential of a Continuous Martingale 606

1e．Convergencein Terms of Hellinger Processes 609

2．Convergence of the Log-Likelihood to a Process with Independent Increments 612

2a．Introduction Statement of the Results 612

2b．The Proof of Theorem 2.12 615

2c．Example：Point Processes 619

3．The Statistical Invariance Principle 620

3a．General Results 621

3b．Convergence to a Gaussian Martingale 623

Bibliographical Comments 629

References 641

Index of Symbols 653

Index of Terminology 655

Index of Topics 659

Index of Conditions for Limit Theorems 661