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