Semimartingales. A Course on Stochastic Processes
The presented monograph is an advanced book on general martingale theory. A reader with good knowledge of probability and discrete time processes will appreciate the deep results contained in the book. Theorems are formulated for quasimartingales, some parts of the book are devoted to Hilbert space valued processes, and the theory is not restricted to continuous square integrable martingales as is usually the case. Due to these facts, the book is a necessity for all researchers working with general stochastic processes. The book consists of two parts. In the first part the general theory of martingales is given. It starts with basic definitions and facts from stochastic processes; filtration, measurability, adapted and predictable processes, stopping time, and decomposition. The next chapters continue with the martingale property. We find many classical results in a quite general context like Doob's inequalities or convergence theorems. The first part is concluded by a chapter devoted to square integrable semimartingales, quadratic variation and Meyer's process. Here we also find the theory of Hilbert space valued martingales and stochastic integrals with respect to them. In the second part, stochastic calculus is the focus. First, the stochastic integral is built and a semimartingale version of Itô transformation theorem is proved. Then, as an application, the Brownian and Poisson processes are considered and changes of probability, as well as Girsanov formula, are presented. The book culminates with a chapter on SDE. This is not a book which I would recommend as an introduction to stochastic processes and martingales. But for any rigorous work in the theory of martingales, namely non-continuous processes, semimartingales and multidimensional martingales, it is the book that should be consulted first. It is not easy to write a concise book on general martingale theory but in my opinion Michel Métivier has done great job.