Introduction to Stochastic Processes

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.

本書的目標是以相對簡潔的方式介紹隨機過程的要素，其中包括馬可夫鏈和隨機分析這兩個最重要的部分。讀者將直接引導進入主題的核心，進一步的細節和額外的材料則留給了豐富的練習部分供進一步閱讀和學習。

在馬可夫鏈的部分，重點是遞歸性。通過使用最小非負解法，我們處理遞歸性和各種類型的遞歸性。這一過程是逐步進行的，從有限狀態空間到可數狀態空間，從離散時間到連續時間。證明方法採用了現代技術，如耦合和對偶方法。一些非常新的結果也包括在內，例如光譜間隙的估計。第一部分的結構和證明與其他現有的馬可夫鏈教材有所不同。

在隨機分析的部分，我們涵蓋了鞅理論和布朗運動，隨機積分和隨機微分方程，重點是一維情況，以及基於半鞅的多維隨機積分和隨機方程。我們在這裡介紹了三個重要的主題：費曼-卡克公式、隨機時間轉換和吉爾薩諾夫轉換。作為概率論在經典數學中的一個重要應用，我們還處理了凸幾何中著名的布倫-明可夫斯基不等式。

本書還介紹了在不同領域中使用的現代概率論，例如MCMC，甚至是確定性領域：凸幾何和數論。它為學生提供了一個從經典馬可夫鏈到現代隨機分析的新而直接的學習路線。