Fundamentals of Stochastic Processes

Author: Ying Jiangang, Jin Mengwei (editors)
Publisher:
Publish Date: 2006-02-01
Features: This book is a graduate-level textbook on stochastic processes. It consists of four chapters, starting with axiomatic probability theory, with a focus on teaching martingales and Markov processes. It introduces conditional expectation, the construction of measures in infinite-dimensional spaces, Markov chains, Poisson measures and Poisson processes, Brownian motion, stochastic integrals of martingales and continuous martingales, It?'s formula, Girsanov's formula, stochastic differential equations, and also covers right Markov processes, Feller processes and Lévy processes, the potential theory of Brownian motion, free theory, and Killing transformations and time transformations of Markov processes. The book is also equipped with a certain number of exercises of varying difficulty to help readers deepen their understanding and inspire critical thinking. This book can be used as a textbook for graduate courses in various mathematical disciplines such as fundamental mathematics, applied mathematics, computational mathematics, operations research and control theory, probability theory and mathematical statistics. It can also be referenced by graduate students in science and engineering, as well as natural science workers and engineering technicians. The theory of stochastic processes developed in the 20th century is an important branch of probability theory. Technically, it is based on measure theory, but it has a very intuitive background, primarily using mathematical methods to describe and study uncertain phenomena observed in nature. Today, stochastic process theory has broad applications in mathematics and many other fields, becoming one of the fundamental tools that mathematicians should master.
This book starts with fundamental probability theory, focusing on the introduction of Markov processes and martingale theory. It is designed to be both rigorous and accessible, covering the main foundational achievements in stochastic processes of the 20th century. While emphasizing the logical rigor of the entire theory, it also highlights the intuitive background and practical applications of the problems. Each chapter is supplemented with a certain number of exercises to help readers better understand and master the ideas and methods of stochastic process theory.
This book starts with fundamental probability theory, focusing on the introduction of Markov processes and martingale theory, covering the main foundational achievements in stochastic processes of the 20th century. Each chapter is equipped with a certain number of exercises.