Probability and Stochastics
The book is an introduction to the modern theory of probability and stochastic processes. It based on the lecture-notes for two-semester, fairly popular, which Prof. Çinlar, a well-known professor and researcher, at Princeton University,firstname.lastname@example.org, has offered for many years.
The first four chapters are in probability theory: measure and integration, probability theory, convergence and conditioning.
The first chapter is a review of measure and integration in the context of the modern literatura on probability and stochastic processes. The second introduces probability spaces as espeila measure spaces.The chapter three is on convergence, routinely classical, and the chapter four in on conditional expectations included Radon-Nikodyn derivatives.
There follows chapters on martingales and stochastics, Poisson random measures, Brownian motion and Markov Processes.
Martingales are introduced in chapter V, the treatment of continuous martingales contain an improvement, achieved through the introductiion of a Doob martingale, a stopped martingale that is uniformly integrable. Two great theorems are considered:martingale characterization of Brownian motion due to Lévy and the characterization of Poisson process due to Watanabe.
Poisson random measures are in Chapter VI, the treatment is from the point of vue of their uses,specially,of Lévy processes. This chapter pays some attention to processes with jumps. The chapter VIII on Brownian motion is mostly on the standard material. Finally, the Chapter IX, on Markov processes, Itô diffusions and jump-diffusions are introduced via stochastic integral equations, as an integral path in a field of Lévy Proceses.
In short, the book is higher recommended. It provides new simple proofs of important results on Probability Theory and Stochastics Processes. In my opinion, it is a stimulating textbook will be for the teaching and research of the materia. A well written text with excellent tools for many instances, in every day language, and then all written precise in mathematical form.