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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.
List 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 Inte
About the author
He Sheng-Wu, Jia-Gang Wang, Jia-an Yan
Summary
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
