Grundlehren der mathematischen Wissenschaften 288 Limit Theorems for Stochastic Processes Bearbeitet von Jean Jacod, Albert N. Shiryaev Neuausgabe 2002. Buch. xx, 664 S. Hardcover ISBN 978 3 540 43932 5 Format (B x L): 15,5 x 23,5 cm Gewicht: 2510 g Weitere Fachgebiete > Mathematik > Stochastik > Wahrscheinlichkeitsrechnung schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, ebooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte.
Table of Contents Chapter I. The General Theory of Stochastic Processes, Semimartingales and Stochastic Integrals... 1 1. Stochastic Basis, Stopping Times, Optional σ -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
XIV Table of Contents 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
Table of Contents XV 2. Martingale Problems and Semimartingales... 151 2a. Formulation of the Problem... 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 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. Σ 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
XVI Table of Contents 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
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 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 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... 428 4a. Convergence of Non-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 X t 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... 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 (X n )... 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