Semimartingale Theory and Stochastic Calculus is a self-contained and comprehensive book that will be valuable for research mathematicians, statisticians, engineers, and students.
Autorentext
He Sheng-Wu, Jia-Gang Wang, Jia-an Yan
Klappentext
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.
Zusammenfassung
Statistical inference carries great significance in model building from both the theoretical and the applications points of view. Its applications to engineering and economic systems, financial economics, and the biological and medical sciences have made statistical inference for stochastic processes a well-recognized and important branch of statistics and probability. The class of semimartingales includes a large class of stochastic processes, including diffusion type processes, point processes, and diffusion type processes with jumps, widely used for stochastic modeling. Until now, however, researchers have had no single reference that collected the research conducted on the asymptotic theory for semimartingales.Semimartingales and their Statistical Inference, fills this need by presenting a comprehensive discussion of the asymptotic theory of semimartingales at a level needed for researchers working in the area of statistical inference for stochastic processes. The author brings together into one volume the state-of-the-art in the inferential aspect for such processes. The topics discussed include:Asymptotic likelihood theoryQuasi-likelihoodLikelihood and efficiencyInference for counting processesInference for semimartingale regression models The author addresses a number of stochastic modeling applications from engineering, economic systems, financial economics, and medical sciences. He also includes some of the new and challenging statistical and probabilistic problems facing today's active researchers working in the area of inference for stochastic processes.
Inhalt
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