Limit Theorems for Stochastic Processes

Transcription

1 Grundlehren der mathematischen Wissenschaften 288 Limit Theorems for Stochastic Processes Bearbeitet von Jean Jacod, Albert N. Shiryaev Neuausgabe Buch. xx, 664 S. Hardcover ISBN 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.

2 Table of Contents Chapter I. The General Theory of Stochastic Processes, Semimartingales and Stochastic Integrals Stochastic Basis, Stopping Times, Optional σ -Field, Martingales a. Stochastic Basis b. Stopping Times c. The Optional σ -Field d. The Localization Procedure e. Martingales f. The Discrete Case Predictable σ -Field, Predictable Times a. The Predictable σ -Field b. Predictable Times c. Totally Inaccessible Stopping Times d. Predictable Projection e. The Discrete Case Increasing Processes a. Basic Properties b. Doob-Meyer Decomposition and Compensators of Increasing Processes c. Lenglart Domination Property d. The Discrete Case Semimartingales and Stochastic Integrals a. Locally Square-Integrable Martingales b. Decompositions of a Local Martingale c. Semimartingales d. Construction of the Stochastic Integral e. Quadratic Variation of a Semimartingale and Ito s Formula f. Doléans-Dade Exponential Formula g. The Discrete Case... 62

3 XIV Table of Contents Chapter II. Characteristics of Semimartingales and Processes with Independent Increments Random Measures a. General Random Measures b. Integer-Valued Random Measures c. A Fundamental Example: Poisson Measures d. Stochastic Integral with Respect to a Random Measure Characteristics of Semimartingales a. Definition of the Characteristics b. Integrability and Characteristics c. A Canonical Representation for Semimartingales d. Characteristics and Exponential Formula Some Examples a. The Discrete Case b. More on the Discrete Case c. The One-Point Point Process and Empirical Processes Semimartingales with Independent Increments a. Wiener Processes b. Poisson Processes and Poisson Random Measures c. Processes with Independent Increments and Semimartingales d. Gaussian Martingales Processes with Independent Increments Which Are Not Semimartingales a. The Results b. The Proofs Processes with Conditionally Independent Increments Progressive Conditional Continuous PIIs Semimartingales, Stochastic Exponential and Stochastic Logarithm a. More About Stochastic Exponential and Stochastic Logarithm b. Multiplicative Decompositions and Exponentially Special Semimartingales Chapter III. Martingale Problems and Changes of Measures Martingale Problems and Point Processes a. General Martingale Problems b. Martingale Problems and Random Measures c. Point Processes and Multivariate Point Processes

4 Table of Contents XV 2. Martingale Problems and Semimartingales a. Formulation of the Problem b. Example: Processes with Independent Increments c. Diffusion Processes and Diffusion Processes with Jumps d. Local Uniqueness Absolutely Continuous Changes of Measures a. The Density Process b. Girsanov s Theorem for Local Martingales c. Girsanoy s Theorem for Random Measures d. Girsanov s Theorem for Semimartingales e. The Discrete Case Representation Theorem for Martingales a. Stochastic Integrals with Respect to a Multi-Dimensional Continuous Local Martingale b. Projection of a Local Martingale on a Random Measure c. The Representation Property d. The Fundamental Representation Theorem Absolutely Continuous Change of Measures: Explicit Computation of the Density Process a. All P-Martingales Have the Representation Property Relative to X b. P Has the Local Uniqueness Property c. Examples Integrals of Vector-Valued Processes and σ martingales a. Stochastic Integrals with Respect to a Multi-Dimensional Locally Square-integrable Martingale b. Integrals with Respect to a Multi-Dimensional Process of Locally Finite Variation c. Stochastic Integrals with Respect to a Multi-Dimensional Semimartingale d. Stochastic Integrals: A Predictable Criterion e. Σ localization and σ martingales Laplace Cumulant Processes and Esscher s Change of Measures a. Laplace Cumulant Processes of Exponentially Special Semimartingales b. Esscher Change of Measure

5 XVI Table of Contents Chapter IV. Hellinger Processes, Absolute Continuity and Singularity of Measures Hellinger Integrals and Hellinger Processes a. Kakutani-Hellinger Distance and Hellinger Integrals b. Hellinger Processes c. Computation of Hellinger Processes in Terms of the Density Processes d. Some Other Processes of Interest e. The Discrete Case Predictable Criteria for Absolute Continuity and Singularity a. Statement of the Results b. The Proofs c. The Discrete Case Hellinger Processes for Solutions of Martingale Problems a. The General Setting b. The Case Where P and P Are Dominated by a Measure Having the Martingale Representation Property c. The Case Where Local Uniqueness Holds Examples a. Point Processes and Multivariate Point Processes b. Generalized Diffusion Processes c. Processes with Independent Increments Chapter V. Contiguity, Entire Separation, Convergence in Variation Contiguity and Entire Separation a. General Facts b. Contiguity and Filtrations Predictable Criteria for Contiguity and Entire Separation a. Statements of the Results b. The Proofs c. The Discrete Case Examples a. Point Processes b. Generalized Diffusion Processes c. Processes with Independent Increments Variation Metric a. Variation Metric and Hellinger Integrals b. Variation Metric and Hellinger Processes

6 Table of Contents XVII 4c. Examples: Point Processes and Multivariate Point Processes d. Example: Generalized Diffusion Processes Chapter VI. Skorokhod Topology and Convergence of Processes The Skorokhod Topology a. Introduction and Notation b. The Skorokhod Topology: Definition and Main Results c. Proof of Theorem Continuity for the Skorokhod Topology a. Continuity Properties of some Functions b. Increasing Functions and the Skorokhod Topology Weak Convergence a. Weak Convergence of Probability Measures b. Application to Càdlàg Processes Criteria for Tightness: The Quasi-Left Continuous Case a. Aldous Criterion for Tightness b. Application to Martingales and Semimartingales Criteria for Tightness: The General Case a. Criteria for Semimartingales b. An Auxiliary Result c. Proof of Theorem Convergence, Quadratic Variation, Stochastic Integrals a. The P-UT Condition b. Tightness and the P-UT Property c. Convergence of Stochastic Integrals and Quadratic Variation d. Some Additional Results Chapter VII. Convergence of Processes with Independent Increments Introduction to Functional Limit Theorems Finite-Dimensional Convergence a. Convergence of Infinitely Divisible Distributions b. Some Lemmas on Characteristic Functions c. Convergence of Rowwise Independent Triangular Arrays d. Finite-Dimensional Convergence of PII-Semimartingales to a PII Without Fixed Time of Discontinuity Functional Convergence and Characteristics a. The Results b. Sufficient Condition for Convergence Under

7 XVIII Table of Contents 3c. Necessary Condition for Convergence d. Sufficient Condition for Convergence More on the General Case a. Convergence of Non-Infinitesimal Rowwise Independent Arrays b. Finite-Dimensional Convergence for General PII c. Another Necessary and Sufficient Condition for Functional Convergence The Central Limit Theorem a. The Lindeberg-Feller Theorem b. Zolotarev s Type Theorems c. Finite-Dimensional Convergence of PII s to a Gaussian Martingale d. Functional Convergence of PII s to a Gaussian Martingale Chapter VIII. Convergence to a Process with Independent Increments Finite-Dimensional Convergence, a General Theorem a. Description of the Setting for This Chapter b. The Basic Theorem c. Remarks and Comments Convergence to a PII Without Fixed Time of Discontinuity a. Finite-Dimensional Convergence b. Functional Convergence c. Application to Triangular Arrays d. Other Conditions for Convergence Applications a. Central Limit Theorem: Necessary and Sufficient Conditions b. Central Limit Theorem: The Martingale Case c. Central Limit Theorem for Triangular Arrays d. Convergence of Point Processes e. Normed Sums of I.I.D. Semimartingales f. Limit Theorems for Functionals of Markov Processes g. Limit Theorems for Stationary Processes Convergence to a General Process with Independent Increments a. Proof of Theorem 4.1 When the Characteristic Function of X t Vanishes Almost Nowhere b. Convergence of Point Processes c. Convergence to a Gaussian Martingale

8 Table of Contents XIX 5. Convergence to a Mixture of PII s, Stable Convergence and Mixing Convergence a. Convergence to a Mixture of PII s b. More on the Convergence to a Mixture of PII s c. Stable Convergence d. Mixing Convergence e. Application to Stationary Processes Chapter IX. Convergence to a Semimartingale Limits of Martingales a. The Bounded Case b. The Unbounded Case Identification of the Limit a. Introductory Remarks b. Identification of the Limit: The Main Result c. Identification of the Limit Via Convergence of the Characteristics d. Application: Existence of Solutions to Some Martingale Problems Limit Theorems for Semimartingales a. Tightness of the Sequence (X n ) b. Limit Theorems: The Bounded Case c. Limit Theorems: The Locally Bounded Case Applications a. Convergence of Diffusion Processes with Jumps b. Convergence of Step Markov Processes to Diffusions c. Empirical Distributions and Brownian Bridge d. Convergence to a Continuous Semimartingale: Necessary and Sufficient Conditions Convergence of Stochastic Integrals a. Characteristics of Stochastic Integrals b. Statement of the Results c. The Proofs Stability for Stochastic Differential Equation a. Auxiliary Results b. Stochastic Differential Equations c. Stability