Limit Theorems for Stochastic Processes

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8 Οκτ 2013 (πριν από 3 χρόνια και 8 μήνες)

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Jean Jacod
Albert N. Shiryaev
Limit Theorems for
Stochastic Processes
Second edition
Springer
Table of Contents
Chapter I. The General Theory of Stochastic Processes,
Semimartingales and Stochastic Integrals 1
1. Stochastic Basis, Stopping Times, Optional a-Field,
Martingales 1
§la. Stochastic Basis 2
§lb. Stopping Times 4
§lc. The Optional CT -Field 5
§ld. The Localization Procedure 8
§le. Martingales 10
§lf. The Discrete Case 13
2. Predictable a-Field, Predictable Times 16
§2a. The Predictable a-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. Doleans-Dade Exponential Formula 58
§4g. The Discrete Case ' 62
XIV Table of Contents
Chapter II. Characteristics of Semimartingales and Processes
with Independent Increments o 64
1. Random Measures 6 4
§la. General Random Measures 65
§lb. Integer-Valued Random Measures 68
§lc. A Fundamental Example: Poisson Measures 70
§ld. 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 9 1
§3a. The Discrete Case 9 1
§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 10 2
§4b. Poisson Processes and Poisson Random Measures 103
§4c. Processes with Independent Increments and Semimartingales .. 106
§4d. Gaussian Martingales I l l
5. Processes with Independent Increments
Which Are Not Semimartingales 114
§5a. The Results 11 4
§5b. The Proofs 11 6
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
§la. General Martingale Problems 143
§lb. Martingale Problems and Random Measures 144
§lc. Point Processes and Multivariate Point Processes 146
Table of Contents XV
2. Martingale Problems and Semimartingales 151
§2a. Formulation of the Probjem 152
§2b. Example: Processes with Independent Increments 154
§2c. Diffusion 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 CT-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 a-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
XVI Table of Contents
Chapter IV. Hellinger Processes, Absolute Continuity
and Singularity of Measures 22 7
1. Hellinger Integrals and Hellinger Processes 228
§la. Kakutani-Hellinger Distance and Hellinger Integrals 228
§lb. Hellinger Processes 23 0
§lc. Computation of Hellinger Processes in Terms
of the Density Processes 234
§ld. Some Other Processes of Interest 237
§le. The Discrete Case 24 2
2. Predictable Criteria for Absolute Continuity and Singularity 245
§2a. Statement of the Results , 245
§2b. The Proofs 24 8
§2c. The Discrete Case 25 2
3. Hellinger Processes for Solutions of Martingale Problems 254
§3a. The General Setting 25 5
§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 27 2
§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
§la. General Facts 28 4
§lb. Contiguity and Filiations 290
2. Predictable Criteria for Contiguity and Entire Separation 291
§2a. Statements of the Results 291
§2b. The Proofs 29 4
§2c. The Discrete Case 30 1
3. Examples 30 4
§3a. Point Processes 30 4
§3b. Generalized Diffusion Processes 305
§3c. Processes with Independent Increments 306
4. Variation Metric 30 9
§4a. Variation Metric and Hellinger Integrals '•. 310
§4b. Variation Metric and Hellinger Processes 312
Table of Contents XVII
§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
§la. Introduction and Notation 325
§lb. The Skorokhod Topology: Definition and Main Results 327
§lc. 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 34 7
§3a. Weak Convergence of Probability Measures 347
§3b. Application to Cadlag 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 PH-Semimartingales
to a PII Without Fixed Time of Discontinuity 408
3. Functional Convergence and Characteristics 413
§3a. The Results 41 4
§3b. Sufficient Condition for Convergence Under 2.48 418
XVIII Table of Contents
§3c. Necessary Condition for Convergence 418
§3d. Sufficient Condition for Convergence 424
4. More on the General Case 42 8
§4a. Convergence of Non-Infinitesimal Rowwise Independent'
Arrays 42 8
§4b. Finite-Dimensional Convergence for General PII 436
§4c. Another Necessary and Sufficient Condition for Functional
Convergence 43 9
5. The Central Limit Theorem 44 4
§5a. The Lindeberg-Feller Theorem 445
§5b. Zolotarev's Type Theorems 446
§5c. Finite-Dimensional Convergence of PII's to a Gaussian
Martingale 45 0
§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
§la. Description of the Setting for This Chapter 456
§lb. The Basic Theorem 45 7
§lc. Remarks and Comments 45 9
2. Convergence to a PII Without Fixed Time of Discontinuity 460
§2a. Finite-Dimensional Convergence 461
§2b. Functional Convergence 46 4
§2c. Application to Triangular Arrays 465
§2d. Other Conditions for Convergence 467
3. Applications 46 9
§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 X,
Vanishes Almost Nowhere 501
§4b. Convergence of Point Processes 503
§4c. Convergence to a Gaussian Martingale . .. ; 504
Table of Contents XIX
5. Convergence to a Mixture of PII's, Stable Convergence
and Mixing Convergence . .. i 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 52 1
§la. The Bounded Case 52 2
§lb. 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 53 5
3. Limit Theorems for Semimartingales 540
§3a. Tightness of the Sequence (X") 541
§3b. Limit Theorems: The Bounded Case 546
§3c. Limit Theorems: The Locally Bounded Case 550
4. Applications 55 4
§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 57 0
6. Stability for Stochastic Differential Equation 575
§6a. Auxiliary Results 57 6
§6b. Stochastic Differential Equations 577
§6c. Stability '. . . . 578
XX Table of Contents
7. Stable Convergence to a Progressive Conditional Continuous PI I.... 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
§la. introduction, Statement of the Main Results 593
§lb. An Auxiliary Computation 597
§lc. Proofs of Theorems 1.12 and 1.16 603
§ld. Convergence to the Exponential of a Continuous Martingale .. . 606
§le. Convergence in 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 62 1
§3b. Convergence to a Gaussian Martingale 623
Bibliographical Comments 629
References 64 1
Index of Symbols 65 3
Index of Terminology 65 5
Index of Topics 65 9
Index of Conditions for Limit Theorems 661