Semimartingale Theory and Stochastic Calculus

 
 
Routledge (Verlag)
  • erschienen am 9. Juli 2019
  • |
  • 400 Seiten
 
E-Book | PDF mit Adobe DRM | Systemvoraussetzungen
978-1-351-41696-2 (ISBN)
 
Semimartingale Theory and Stochastic Calculus presents a systematic and detailed account of the general theory of stochastic processes, the semimartingale theory, and related stochastic calculus. The book emphasizes stochastic integration for semimartingales, characteristics of semimartingales, predictable representation properties and weak convergence of semimartingales. It also includes a concise treatment of absolute continuity and singularity, contiguity, and entire separation of measures by semimartingale approach. Two basic types of processes frequently encountered in applied probability and statistics are highlighted: processes with independent increments and marked point processes encountered frequently in applied probability and statistics.

Semimartingale Theory and Stochastic Calculus is a self-contained and comprehensive book that will be valuable for research mathematicians, statisticians, engineers, and students.
  • Englisch
  • Boca Raton
  • |
  • USA
Taylor & Francis Ltd
  • Für höhere Schule und Studium
978-1-351-41696-2 (9781351416962)
weitere Ausgaben werden ermittelt
He Sheng-Wu, Jia-Gang Wang, Jia-an Yan
  • Cover
  • Half Title
  • Title Page
  • Copyright Page
  • Table of Contents
  • Chapter I: Preliminaries
  • §1. Monotone Class Theorems
  • §2. Uniform Integrability
  • §3. Essential Suprema
  • §4. The Generalization of Conditional Expectation
  • §5. Analytic Sets and Choquet Capacity
  • §6. Lebesgue-Stieltjes Integrals
  • Problems and Complements
  • Chapter II: Classical Martingale Theory
  • §1. Elementary Inequalities
  • §2. Convergence Theorems
  • §3. Decomposition Theorems for Supermartingales
  • §4. Doob's Stopping Theorem
  • §5. Martingales with Continuous Time
  • §6. Processes with Independent Increments
  • Problems and Complements
  • Chapter III: Processes and Stopping Times
  • § 1. Stopping Times
  • §2. Progressively Measurable, Optional and Predictable Processes
  • §3. Predictable and Accessible Times
  • §4. Processes with Finite Variation
  • §5. Changes of Time
  • Problems and Complements
  • Chapter IV: Section Theorems and Their Applications
  • §1. Section Theorems
  • §2. a.s. Foretellability of Predicatable Times
  • §3. Totally Inaccessible Times
  • §4. Complete Filtrations and the Usual Conditions
  • §5. Applications to Martingales
  • Problems and Complements
  • Chapter V: Projections of Processes
  • §1. Projections of Measurable Processes
  • §2. Dual Projections of Increasing Processes
  • §3. Applications to Stopping Times and Processes
  • §4. Doob-Meyer Decomposition Theorem
  • §5. Filtrations of Discrete Type
  • Problems and Complements
  • Chapter VI: Martingales with Integrable Variation and Square Integrable Martingales
  • §1. Martingales with Integrable Variation
  • §2. Square Integrable Martingales
  • §3. The Structure of Purely Discontinuous Square Integrable Martingales
  • §4. Quadratic Variation
  • Problems and Complements
  • Chapter VII: Local Martingales
  • §1. The Localization of Classes of Processes
  • §2. The Decomposition of Local Martingales
  • §3. The Characterization of Jumps of Local Martingales
  • Problems and Complements
  • Chapter VIII: Semimartingales and Quasimartingales
  • §1. Semimartingales and Special Semimartingales
  • §2. Quasimartingales and Their Rao Decompositions
  • §3. Semimartingales on Stochastic Sets of Interval Type
  • §4. Convergence Theorems for Semimartingales
  • Problems and Complements
  • Chapter IX: Stochastic Integrals
  • §1. Stochastic Integrals of Predictable Processes with Respect to Local Martingales
  • §2. Compensated Stochastic Integrals of Progressive Processes with Respect to Local Martingales
  • §3. Stochastic Integrals of Predictable Processes with Respect to Semimartingales
  • §4. Lenglart's Inequality and Convergence Theorem for Stochastic Integrals
  • §5. Itô Formula and Doléans-Dade Exponential Formula
  • §6. Local Times of Semimartingales
  • §7. Stochastic Differential Equations: Métivier-Pellaumail's Method
  • Problems and Complements
  • Chapter X: Martingale Spaces H1. and BMO
  • §1. H1-martingales and BMO-Martingales
  • §2. Fefferman's Inequality
  • §3. The Dual Space of H1
  • §4. Davis Inequalities
  • §5. Burkholder-Davis-Gundy Inequality
  • §6. Martingale Space HP, p> 1
  • §7. John-Nirenberg Inequality
  • Problems and Complements
  • Chapter XI: The Characteristics of Semimartingales
  • §1. Random Measures
  • §2. The Integral Representation of Semimartingales
  • §3. Lévy Processes
  • §4. Step Processes
  • Problems and Complements
  • Chapter XII: Changes of Measures
  • §1. Local Absolute Continuity
  • §2. Girsanov's Theorems for Local Martingales and Semimartingales
  • §3. Girsanov's Theorems for Random Measures
  • §4. The Characterization for Semimartingales
  • Problems and Complements
  • Chapter XIII: Predictable Representaton Property
  • §1. The Strong Property of Predictable Representation
  • §2. The Weak Property of Predictable Representation
  • §3. The Relation between Two Kinds of Predictable Representation Properties
  • §4. The Predictable Representation Property of Lévy Processes
  • Problems and Complements
  • Chapter XIV: Absolute Continuity and Contiguity of Measures
  • §1. Hellinger Processes
  • §2. Absolute Continuity and Singularity
  • §3. Contiguity, Entire Separation and Convergence in Variation
  • §4. Measures Induced by Lévy Processes
  • Problems and Complements
  • Chapter XV: Weak Convergence for Cadlag Processes
  • §1. D[0, oo[ and Skorokhod Topology
  • §2. Continuity for Skorokhod Topology
  • §3. Weak Convergence and Tightness
  • §4. Weak Convergence of Step Processes
  • Problems and Complements
  • Chapter XVI: Weak Convergence for Semimartingales
  • §1. Convergence to a Quasi-left-continuous Semimartingale
  • §2. Convergence to a Lévy Process
  • §3. Convergence to a Continuous Lévy process
  • §4. Convergence to a Generalized Diffusion
  • Problems and Complements
  • References
  • Index

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