This is a brief introduction to stochastic processes studying certain elementary continuous-time processes. After a description of the Poisson process and related processes with independent increments as well as a brief look at Markov processes with a finite number of jumps, the author proceeds to introduce Brownian motion and to develop stochastic integrals and Ità ́'s theory in the context of one-dimensional diffusion processes. The book ends with a brief survey of the general theory of Markov processes. The book is based on courses given by the author at the Courant Institute and can be used as a sequel to the author's successful book Probability Theory in this series. Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University. This Is A Brief Introduction To Stochastic Processes Studying Certain Elementary Continuous-time Processes. After A Description Of The Poisson Process And Related Processes With Independent Increments As Well As A Brief Look At Markov Processes With A Finite Number Of Jumps, The Author Proceeds To Introduce Brownian Motion And To Develop Stochastic Integrals And Ito's Theory In The Context Of One-dimensional Diffusion Processes. The Book Ends With A Brief Survey Of The General Theory Of Markov Processes. The Book Is Based On Courses Given By The Author At The Courant Institute And Can Be Used As A Sequel To The Author's Successful Book Probability Theory In This Series.--jacket. Ch. 1. Introduction -- 1.1. Continuous Time Processes -- 1.2. Continuous Parameter Martingales -- 1.3. Semimartingales -- 1.4. Martingales And Stochastic Integrals -- Ch. 2. Processes With Independent Increments -- 2.1. The Basic Poisson Process -- 2.2. Compound Poisson Processes -- 2.3. Infinite Number Of Small Jumps -- 2.4. Infinitesimal Generators -- 2.5. Some Associated Martingales -- Ch. 3. Poisson Point Processes -- 3.1. Point Processes -- 3.2. Poisson Point Process -- Ch. 4. Jump Markov Processes -- 4.1. Simple Examples -- 4.2. Semigroups Of Operators -- 4.3. Example: Birth And Death Processes -- 4.4. Markov Processes And Martingales -- 4.5. Explosion -- 4.6. Recurrence And Transience. S.r.s. Varadhan. Includes Bibliographical References And Index. This is a brief introduction to stochastic processes studying certain elementary continuous-time processes. After a description of the Poisson process and related processes with independent increments as well as a brief look at Markov processes with a finite number of jumps, the author proceeds to introduce Brownian motion and to develop stochastic integrals and Itô's theory in the context of one-dimensional diffusion processes. The book ends with a brief survey of the general theory of Markov processes. The book is based on courses given by the author at the Courant Institute and can be used as a sequel to the author's successful book Probability Theory in this series. Contents 8 Preface 10 CHAPTER 1 Introduction 12 CHAPTER 2 Processes with Independent Increments 24 CHAPTER 3 Poisson Point Processes 36 CHAPTER 4 Jump Markov Processe s 40 CHAPTER 5 Brownian Motion 60 CHAPTER 6 One-Dimensional Diffusions 98 CHAPTER 7 General Theory of Markov Processes 118 APPENDIX A Measures on Polish Spaces 124 APPENDIX B Additional Remarks 132 Bibliography 134 Index 136
