随机过程用的极限定理 第2版 英文PDF|Epub|txt|kindle电子书版本网盘下载

随机过程用的极限定理 第2版 英文PDF格式电子书版下载

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Chapter Ⅰ．The General Theory of Stochastic Processes,Semimartingales and Stochastic Integrals1

1．Stochastic Basis,Stopping Times,Optionalσ-Field,Martingales1

1a．Stochastic Basis2

1b．Stopping Times4

1c．The Optional σ-Field5

1d．The Localization Procedure8

1e．Martingales10

1f．The Discrete Case13

2．Predictable σ-Field,Predictable Times16

2a．The Predictable σ-Field16

2b．Predictable Times17

2c．Totally Inaccessible Stopping Times20

2d．Predictable Projection22

2e．The Discrete Case25

3．Increasing Processes27

3a．Basic Properties27

3b．Doob-Meyer Decomposition and Compensators of Increasing Processes32

3c．Lenglart Domination Property35

3d．The Discrete Case36

4．Semimartingales and Stochastic Integrals38

4a．Locally Square-Integrable Martingales38

4b．Decompositions of a Local Martingale40

4c．Semimartingales43

4d．Construction of the Stochastic Integral46

4e．Quadratic Variation of a Semimartingale and Ito's Formula51

4f．Doléans-Dade Exponential Formula58

4g．The Discrete Case62

Chapter Ⅱ．Characteristics of Semimartingales and Processes with Independent Increments64

1．Random Measures64

1a．General Random Measures65

1b．Integer-Valued Random Measures68

1c．A Fundamental Example：Poisson Measures70

1d．Stochastic Integral with Respect to a Random Measure71

2．Characteristics of Semimartingales75

2a．Definition of the Characteristics75

2b．Integrability and Characteristics81

2c．A Canonical Representation for Semimartingales84

2d．Characteristics and Exponential Formula85

3．Some Examples91

3a．The Discrete Case91

3b．More on the Discrete Case93

3c．The"One-Point"Point Process and Empirical Processes97

4．Semimartingales with Independent Increments101

4a．Wiener Processes102

4b．Poisson Processes and Poisson Random Measures103

4c．Processes with Independent Increments and Semimartingales106

4d．Gaussian Martingales111

5．Processes with Independent Increments Which Are Not Semimartingales114

5a．The Results114

5b．The Proofs116

6．Processes with Conditionally Independent Increments124

7．Progressive Conditional Continuous PIIs128

8．Semimartingales,Stochastic Exponential and Stochastic Logarithm134

8a．More About Stochastic Exponential and Stochastic Logarithm134

8b．Multiplicative Decompositions and Exponentially Special Semimartingales138

Chapter Ⅲ．Martingale Problems and Changes of Measures142

1．Martingale Problems and Point Processes143

1a．General Martingale Problems143

1b．Martingale Problems and Random Measures144

1c．Point Processes and Multivariate Point Processes146

2．Martingale Problems and Semimartingales151

2a．Formulation of the Problem152

2b．Example：Processes with Independent Increments154

2c．Diflusion Processes and Diffusion Processes with Jumps155

2d．Local Uniqueness159

3．Absolutely Continuous Changes of Measures165

3a．The Density Process165

3b．Girsanov's Theorem for Local Martingales168

3c．Girsanoy's Theorem for Random Measures170

3d．Girsanov's Theorem for Semimartingales172

3e．The Discrete Case177

4．Representation Theorem for Martingales179

4a．Stochastic Integrals with Respect to a Multi-Dimensional Continuous Local Martingale179

4b．Projection of a Local Martingale on a Random Measure182

4c．The Representation Property185

4d．The Fundamental Representation Theorem187

5．Absolutely Continuous Change of Measures：Explicit Computation of the Density Process191

5a．All P-Martingales Have the Representation Property Relative to X192

5b．P′Has the Local Uniqueness Property196

5c．Examples200

6．Integrals of Vector-Valued Processes and σ-martingales203

6a．Stochastic Integrals with Respect to a Multi-Dimensional Locally Square-integrable Martingale204

6b．Integrals with Respect to a Multi-Dimensional Process of Locally Finite Variation206

6c．Stochastic Integrals with Respect to a Multi-Dimensional Semimartingale207

6d．Stochastic Integrals：A Predictable Criterion212

6e．∑-localization and σ-martingales214

7．Laplace Cumulant Processes and Esscher's Change of Measures219

7a．Laplace Cumulant Processes of Exponentially Special Semimartingales219

7b．Esscher Change of Measure222

Chapter Ⅳ．Hellinger Processes,Absolute Continuity and Singularity of Measures227

1．Hellinger Integrals and Hellinger Processes228

1a．Kakutani-Hellinger Distance and Hellinger Integrals228

1b．Hellinger Processes230

1c．Computation of Hellinger Processes in Terms of the Density Processes234

1d．Some Other Processes of Interest237

1e．The Discrete Case242

2．Predictable Criteria for Absolute Continuity and Singularity245

2a．Statement of the Results245

2b．The Proofs248

2c．The Discrete Case252

3．Hellinger Processes for Solutions of Martingale Problems254

3a．The General Setting255

3b．The Case Where P and P′Are Dominated by a Measure Having the Martingale Representation Property257

3c．The Case Where Local Uniqueness Holds266

4．Examples272

4a．Point Processes and Multivariate Point Processes272

4b．Generalized Diffusion Processes275

4c．Processes with Independent Increments277

Chapter Ⅴ．Contiguity,Entire Separation,Convergence in Variation284

1．Contiguity and Entire Separation284

1a．General Facts284

1b．Contiguity and Filtrations290

2．Predictable Criteria for Contiguity and Entire Separation291

2a．Statements of the Results291

2b．The Proofs294

2c．The Discrete Case301

3．Examples304

3a．Point Processes304

3b．Generalized Diffusion Processes305

3c．Processes with Independent Increments306

4．Variation Metric309

4a．Variation Metric and Hellinger Integrals310

4b．Variation Metric and Hellinger Processes312

4c．Examples：Point Processes and Multivariate Point Processes318

4d．Example：Generalized Diffusion Processes322

Chapter Ⅵ．Skorokhod Topology and Convergence of Processes324

1．The Skorokhod Topology325

1a．Introduction and Notation325

1b．The Skorokhod Topology：Definition and Main Results327

1c．Proof of Theorem 1.14329

2．Continuity for the Skorokhod Topology337

2a．Continuity Properties of some Functions337

2b．Increasing Functions and the Skorokhod Topology342

3．Weak Convergence347

3a．Weak Convergence of Probability Measures347

3b．Application to Càdlàg Processes348

4．Criteria for Tightness：The Quasi-Left Continuous Case355

4a．Aldous'Criterion for Tightness356

4b．Application to Martingales and Semimartingales358

5．Criteria for Tightness：The General Case362

5a．Criteria for Semimartingales362

5b．An Auxiliary Result365

5c．Proof of Theorem 5.17367

6．Convergence,Quadratic Variation,Stochastic Integrals376

6a．The P-UT Condition377

6b．Tightness and the P-UT Property382

6c．Convergence of Stochastic Integrals and Quadratic Variation382

6d．Some Additional Results386

Chapter Ⅶ．Convergence of Processes with Independent Increments389

1．Introduction to Functional Limit Theorems390

2．Finite-Dimensional Convergence394

2a．Convergence of Infinitely Divisible Distributions394

2b．Some Lemmas on Characteristic Functions398

2c．Convergence of Rowwise Independent Triangular Arrays402

2d．Finite-Dimensional Convergence of PII-Semimartingales to a PII Without Fixed Time of Discontinuity408

3．Functional Convergence and Characteristics413

3a．The Results414

3b．Sufficient Condition for Convergence Under2.48418

3c．Necessary Condition for Convergence418

3d．Sufficient Condition for Convergence424

4．More on the General Case428

4a．Convergence ofNon-Infinitesimal Rowwise Independent　Arrays428

4b．Finite-Dimensional Convergence for General PII436

4c．Another Necessary and Sufficient Condition for Functional Convergence439

5．The Central Limit Theorem444

5a．The Lindeberg-Feller Theorem445

5b．Zolotarev's Type Theorems446

5c．Finite-Dimensional Convergence of PII's to a Gaussian Martingale450

5d．Functional Convergence of PII's to a Gaussian Martingale452

Chapter Ⅷ．Convergence to a Process with Independent Increments456

1．Finite-Dimensional Convergence,a General Theorem456

1a．Description of the Setting for This Chapter456

1b．The Basic Theorem457

1c．Remarks and Comments459

2．Convergence to a PII Without Fixed Time of Discontinuity460

2a．Finite-Dimensional Convergence461

2b．Functional Convergence464

2c．Application to Triangular Arrays465

2d．Other Conditions for Convergence467

3．Applications469

3a．Central Limit Theorem：Necessary and Sufficient Conditions470

3b．Central Limit Theorem：The Martingale Case473

3c．Central Limit Theorem for Triangular Arrays477

3d．Convergence of Point Processes478

3e．Normed Sums of I.I.D.Semimartingales481

3f．Limit Theorems for Functionals of Markov Processes486

3g．Limit Theorems for Stationary Processes489

4．Convergence to a General Process with Independent Increments499

4a．Proof of Theorem 4.1 When the Characteristic Function of Xt Vanishes Almost Nowhere501

4b．Convergence of Point Processes503

4c．Convergence to a Gaussian Martingale504

5．Convergence to a Mixture of PII's,Stable Convergence and Mixing Convergence506

5a．Convergence to a Mixture of PII's506

5b．More on the Convergence to a Mixture of PII's510

5c．Stable Convergence512

5d．Mixing Convergence518

5e．Application to Stationary Processes519

Chapter Ⅸ．Convergence to a Semimartingale521

1．Limits of Martingales521

1a．The Bounded Case522

1b．The Unbounded Case524

2．Identification of the Limit527

2a．Introductory Remarks527

2b．Identification of the Limit：The Main Result530

2c．Identification of the Limit Via Convergence of the Characteristics533

2d．Application：Existence of Solutions to Some Martingale Problems535

3．Limit Theorems for Semimartingales540

3a．Tightness of the Sequence(Xn)541

3b．Limit Theorems：The Bounded Case546

3c．Limit Theorems：The Locally Bounded Case550

4．Applications554

4a．Convergence of Diffusion Processes with Jumps554

4b．Convergence of Step Markov Processes to Diffusions557

4c．Empirical Distributions and Brownian Bridge560

4d．Convergence to a Continuous Semimartingale：Necessary and Sufficient Conditions561

5．Convergence of Stochastic Integrals564

5a．Characteristics of Stochastic Integrals564

5b．Statement of the Results567

5c．The Proofs570

6．Stability for Stochastic Differential Equation575

6a．Auxiliary Results576

6b．Stochastic Differential Equations577

6c．Stability578

7．Stable Convergence to a Progressive Conditional Continuous PII583

7a．A General Result583

7b．Convergence of Discretized Processes589

Chapter Ⅹ．Limit Theorems,Density Processes and Contiguity592

1．Convergence of the Density Processes to a Continuous Process593

1a．Introduction,Statement of the Main Results593

1b．An Auxiliary Computation597

1c．Proofs of Theorems 1.12 and 1.16603

1d．Convergence to the Exponential of a Continuous Martingale606

1e．Convergencein Terms of Hellinger Processes609

2．Convergence of the Log-Likelihood to a Process with Independent Increments612

2a．Introduction Statement of the Results612

2b．The Proof of Theorem 2.12615

2c．Example：Point Processes619

3．The Statistical Invariance Principle620

3a．General Results621

3b．Convergence to a Gaussian Martingale623

Bibliographical Comments629

References641

Index of Symbols653

Index of Terminology655

Index of Topics659

Index of Conditions for Limit Theorems661