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.
Table of Contents
PRELIMINARIES. Monotone Class Theorems. Uniform Integrability. Essential Supremum. The Generalization of Conditional Expectation. Analytic Sets and Choquet Capacity. Lebesgue-Stieltjes Integrals. CLASSICAL MARTINGALE THEORY. Elementary Inequalities. Convergence Theorems. Decomposition Theorems for Supermartingales. Doob's Stopping Theorem. Martingales with Continuous Time. Processes with Independent Increments. PROCESSES AND STOPPING TIMES. Stopping Times. Progressive Measurable, Optional and Predictable Processes. Predictable and Accessible Times. Processes with Finite Variation. Changes of Time. SECTION THEROREMS AND THEIR APPLICATIONS. Section Theorems. A.s. Foretellability of Predicatable Times. Totally Inaccessible Times. Complete Filtrations and the Usual Conditions. Applications to Martingales. PROJECTIONS OF PROCESSES. Projections of Measurable Processes. Dual Projections of Increasing Processes. Applications to Stopping Times and Processes. Doob-Meyer Decomposition Theorem. Filtrations of Discrete Type. MARTINGALES WITH INTEGRABLE VARIATION AND SQUARE INTEGRABLE MARTINGALES. Martingales with Integrable Variation. Stable Subspaces of Square Integrable Martingales. The Structure of Purely Discontinuous Square Integrable Martingales. Quadratic Variation. LOCAL MARTINGALES. The Localization of Classes of Processes. The Decomposition of Local Martingales. The Characterization of Jumps of Local Martingales. SEMIMARTINGALES AND QUASIMARTINGALES. Semimartingales and Special Semimartingales. Quasimartingales and Their Rao Decompositions. Semimartingales on Stochastic Sets of Interval Type. Convergence Theorems for Semimartingales. STOCHASTIC INTEGRALS. Stochastic Integrals of Predictable Processes with Respect to Local Martingales. Compensated Stochastic Integrals of Progressive Processes with Respect to Local Martingales. Stochastic Integrals of Predictable Processes with Respect to Semimartingales. Lenglart's Inequality and Convergence Theorems for Stochastic Integrals. Ito's Formula and Doleans-Dade Exponential Formula. Local Times of Semimartingales. Stochastic Differential Equations: Metivier-Pellaumail Method. MARTINGALE SPACES H1 AND BMO. H1-Martingales and BMO-Martingales. Fefferman's Inequality. The Dual Space of H1. Davis Inequalities. Burkholder-Davis-Gundy Inequality. Martingale Spaces HR,p > 1. John-Nirenberg Inequality. THE CHARACTERISTICS OF SEMIMARTINGALES. Random Measures. The Integral Representation of Semimartingales. Levy Processes. Jump Processes. CHANGES OF MEASURES. Local Absolute Continuity. Girsanov's Theorems for Local Martingales and Semimartingales. Girsano's Theorems for Random Measures. Semimartingales under Reduction of Filtration and Semimartingale Problem. The Characterization of Semimartingales. PREDICTABLE REPRESENTATION PROPERTY. Strong Predictable Representation Property. Weak Predictable Representation Property. The Relation between the Two Kinds of Predictable Representation Properties. ABSOLUTE CONTINUITY AND CONTIGUITY OF MEASURES. Absolute Continuity and Contiguity of Measures. Hellinger Processes. Predictable Criteria for Absolute Continuity and Contiguity. Absolute Continuity and Contiguity of Semimartingale Measures. WEAK CONVERGENCE OF PROCESSES. D([O,8[) and the Skorokhod Topology. Continuity under the Skorokhod Topology. Weak Convergence of Measures. Criteria for Tightness. Weak Convergence of Jump Processes. WEAK CONVERGENCE OF SEMIMARTINGALES. The Conditions for Weak Convergence to a Quasi-Left-Continuous Semimartingale. The Conditions for Weak Convergence to a Quasi-Left-Continuous Process with Independent Increments. Weak Convergence to a Brownian Motion. Other Applications and Complements.