Stochastic Analysis and Diffusion Processes (Paperback)

Stochastic Analysis and Diffusion Processes presents a simple, mathematical introduction to Stochastic Calculus and its applications. The book builds the basic theory and offers a careful account of important research directions in Stochastic Analysis. The breadth and power of Stochastic Analysis, and probabilistic behavior of diffusion processes are told without compromising on the mathematical details.

Starting with the construction of stochastic processes, the book introduces Brownian motion and martingales. The book proceeds to construct stochastic integrals, establish the Ito formula, and discuss its applications. Next, attention is focused on stochastic differential equations (SDEs) which arise in modeling physical phenomena, perturbed by random forces. Diffusion processes are solutions of SDEs and form the main theme of this book.

The Stroock-Varadhan martingale problem, the connection between diffusion processes and partial differential equations, Gaussian solutions of SDEs, and Markov processes with jumps are presented in successive chapters. The book culminates with a careful treatment of important research topics such as invariant measures, ergodic behavior, and large deviation principle for diffusions.

Examples are given throughout the book to illustrate concepts and results. In addition, exercises are given at the end of each chapter that will help the reader to understand the concepts better. The book is written for graduate students, young researchers and applied scientists who are interested in stochastic processes and their applications. The reader is assumed to be familiar with probability theory at graduate level. The book can be used as a text for a graduate course on Stochastic Analysis.

《隨機分析與擴散過程》是一本簡單且數學化的介紹隨機微積分及其應用的書籍。該書建立了基本理論，並詳細介紹了隨機分析中的重要研究方向。在講述隨機分析的廣度和能力以及擴散過程的概率行為時，不會忽略數學細節。

從隨機過程的構建開始，本書介紹了布朗運動和鞅。接著，本書建立了隨機積分，確立了伊藤公式並討論了其應用。接下來，重點放在隨機微分方程（SDEs）上，這些方程用於建模受隨機力干擾的物理現象。擴散過程是SDEs的解，也是本書的主題。

隨後的章節介紹了Stroock-Varadhan鞅問題、擴散過程與偏微分方程之間的聯繫、SDEs的高斯解以及具有跳躍的馬爾可夫過程。本書以對不變測度、遞歸行為和擴散的大偏差原理等重要研究主題進行了仔細的探討。

本書中穿插了示例以說明概念和結果。此外，每章末尾還提供了練習題，有助於讀者更好地理解概念。本書適用於對隨機過程及其應用感興趣的研究生、年輕研究人員和應用科學家。讀者應該對研究生水平的概率論有所了解。本書可作為隨機分析研究生課程的教材。