Diffusion Processes, Jump Processes, and Stochastic Differential Equations provides a compact exposition of the results explaining interrelations between diffusion stochastic processes, stochastic differential equations and the fractional infinitesimal operators. The draft of this book has been extensively classroom tested by the author at Case Western Reserve University in a course that enrolled seniors and graduate students majoring in mathematics, statistics, engineering, physics, chemistry, economics and mathematical finance. The last topic proved to be particularly popular among students looking for careers on Wall Street and in research organizations devoted to financial problems. Features Quickly and concisely builds from basic probability theory to advanced topics Suitable as a primary text for an advanced course in diffusion processes and stochastic differential equations Useful as supplementary reading across a range of topics.

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.

Stochastic Differential Equations and Diffusion Processes

During the weekend of March 16-18, 1990 the University of North Carolina at Charlotte hosted a conference on the subject of stochastic flows, as part of a Special Activity Month in the Department of Mathematics. This conference was supported jointly by a National Science Foundation grant and by the University of North Carolina at Charlotte. Originally conceived as a regional conference for researchers in the Southeastern United States, the conference eventually drew participation from both coasts of the U. S. and from abroad. This broad-based par ticipation reflects a growing interest in the viewpoint of stochastic flows, particularly in probability theory and more generally in mathematics as a whole. While the theory of deterministic flows can be considered classical, the stochastic counterpart has only been developed in the past decade, through the efforts of Harris, Kunita, Elworthy, Baxendale and others. Much of this work was done in close connection with the theory of diffusion processes, where dynamical systems implicitly enter probability theory by means of stochastic differential equations. In this regard, the Charlotte conference served as a natural outgrowth of the Conference on Diffusion Processes, held at Northwestern University, Evanston Illinois in October 1989, the proceedings of which has now been published as Volume I of the current series. Due to this natural flow of ideas, and with the assistance and support of the Editorial Board, it was decided to organize the present two-volume effort.

This monograph presents a modern treatment of (1) stochastic differential equations and (2) diffusion and jump-diffusion processes. The simultaneous treatment of diffusion processes and jump processes in this book is unique: Each chapter starts from continuous processes and then proceeds to processes with jumps.In the first part of the book, it is shown that solutions of stochastic differential equations define stochastic flows of diffeomorphisms. Then, the relation between stochastic flows and heat equations is discussed. The latter part investigates fundamental solutions of these heat equations (heat kernels) through the study of the Malliavin calculus. The author obtains smooth densities for transition functions of various types of diffusions and jump-diffusions and shows that these density functions are fundamental solutions for various types of heat equations and backward heat equations. Thus, in this book fundamental solutions for heat equations and backward heat equations are constructed independently of the theory of partial differential equations.Researchers and graduate student in probability theory will find this book very useful.

This monograph presents a modern treatment of (1) stochastic differential equations and (2) diffusion and jump-diffusion processes. The simultaneous treatment of diffusion processes and jump processes in this book is unique: Each chapter starts from continuous processes and then proceeds to processes with jumps. In the first part of the book, it is shown that solutions of stochastic differential equations define stochastic flows of diffeomorphisms. Then, the relation between stochastic flows and heat equations is discussed. The latter part investigates fundamental solutions of these heat equations (heat kernels) through the study of the Malliavin calculus. The author obtains smooth densities for transition functions of various types of diffusions and jump-diffusions and shows that these density functions are fundamental solutions for various types of heat equations and backward heat equations. Thus, in this book fundamental solutions for heat equations and backward heat equations are constructed independently of the theory of partial differential equations. Researchers and graduate student in probability theory will find this book very useful.