Stochastic Analysis and Diffusion Processes

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 Itô 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.

Autorentext

Gopinath Kallianpur, Professor Emeritus at University of North Carolina at Chapel Hill, has worked extensively on Stochastic Analysis and is a world renowned expert on stochastic filtering theory. He is the author of Stochastic Filtering Theory, and a co-author of White Noise Theory of Prediction, Filtering and Smoothing, Introduction to Option Pricing Theory, and Stochastic Differential Equations in Infinite Dimensions. P. Sundar is a Professor of Mathematics at Louisiana State University. He works on Stochastic Analysis, and is on the Editorial Board for the journal Communications on Stochastic Analysis. He has co-edited a book titled Infinite Dimensional Stochastic Analysis.

Inhalt

1 Introduction to Stochastic Processes; 2 Brownian Motion and Wiener Measure; 3 Elements of Martingale Theory; 4 Analytic Tools for Brownian Motion; 5 Stochastic Integration; 6 Stochastic Differential Equations; 7 The Martingale Problem; 8 Probability Theory and Partial Differential Equations; 9 Gaussian Solutions; 10 Jump Markov Processes; 11 Invariant Measures and Ergodicity; 12 Large Deviations for Diffusions