Introduction to Stochastic Processes eBook

The objective of this book is to introduce the elements of stochastic processes in a rather concise manner where we present the two most important parts — Markov chains and stochastic analysis. The readers are led directly to the core of the main topics to be treated in the context. Further details and additional materials are left to a section containing abundant exercises for further reading and studying.In the part on Markov chains, the focus is on the ergodicity. By using the minimal nonnegative solution method, we deal with the recurrence and various types of ergodicity. This is done step by step, from finite state spaces to denumerable state spaces, and from discrete time to continuous time. The methods of proofs adopt modern techniques, such as coupling and duality methods. Some very new results are included, such as the estimate of the spectral gap. The structure and proofs in the first part are rather different from other existing textbooks on Markov chains.In the part on stochastic analysis, we cover the martingale theory and Brownian motions, the stochastic integral and stochastic differential equations with emphasis on one dimension, and the multidimensional stochastic integral and stochastic equation based on semimartingales. We introduce three important topics here: the Feynman–Kac formula, random time transform and Girsanov transform. As an essential application of the probability theory in classical mathematics, we also deal with the famous Brunn–Minkowski inequality in convex geometry.This book also features modern probability theory that is used in different fields, such as MCMC, or even deterministic areas: convex geometry and number theory. It provides a new and direct routine for students going through the classical Markov chains to the modern stochastic analysis.Contents:

• Preface to the English Edition
• Preface to the Chinese Edition
• Markov Processes:
  • Discrete-Time Markov Chains
  • Continuous-Time Markov Chains
  • Reversible Markov Chains
  • General Markov Processes
• Stochastic Analysis:
  • Martingale
  • Brownian Motion
  • Stochastic Integral and Diffusion Processes
  • Semimartingale and Stochastic Integral
• Notes
• Bibliography
• Index

Readership: Advanced undergraduate and graduate students in stochastic processes dealing with Markov chains and stochastic analysis.Markov Chain;General Markov Process;Brownian Motion;Martingale;Stochastic Integral;Diffusion Process;Stochastic Differential Equation0Key Features:

• The book is so concise to cover the most important parts in stochastic processes: Markov chains and stochastic analysis
• Some modern and new materials are included, such as the estimation of the first non-trivial eigenvalue and the Brunn–Minkowski inequality
• The book provides abundant exercises for students, regarded as supplements to the main body of the book as well