Semimartingale theory and stochastic calculus (Q2784990)

The Chapter 2 of the handbook under review is an overview of the semimartingale theory and stochastic calculus. It consists of three sections. In Section 1 the main results of the classical theory of martingales, the general theory of stochastic processes and modern martingale theory are presented. In particular, in this section one can found facts concerning optional and predictable processes, projections of measurable processes, dual projections of finite variation processes, Doob-Meyer's decomposition, martingales with integrable variation and uniformly square integrable martingales, local martingales and semimartingales, martingales spaces. In Section 2 the stochastic integrals of real-valued and vector-valued local martingales and semimartingales, for both predictable and progressive integrands, are presented. Among other things, an Itô's formula, the Doléans exponential formula, Tanaka-Meyer's formula for local times of semimartingales, the Fisk-Stratonovich integral, and the Itô stochastic differential equation are also presented. In Section 3 stochastic integration with respect to random measures, characteristics of semimartingales, calculus on Lévy processes, Girsanov's theorems, sufficient conditions for the uniform integrability of exponential martingales, characterization theorem for semimartingales, martingale representation theorems are given.NEWLINENEWLINEFor the entire collection see [Zbl 0979.00017].