A First Look at Stochastic Processes PDF
Preview A First Look at Stochastic Processes
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. A FIRST LOOK AT STOCHASTIC PROCESSES Copyright © 2020 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-120-790-7 ISBN 978-981-120-897-3 (pbk) For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/11488#t=suppl Printed in Singapore Preface This book describes the mathematical theory of stochastic processes, i.e. of quantities that proceed randomly as a function of time. A main example is Markov chains, which are the focus of the first half of the book and also make frequent appearances in the second half. Simple rules about how processes proceed at each step lead to many surprising, interesting, and elegant theorems about what happens in the long run. I have tried to communicate the excitement and intrigue of this subject without requiring extensive background knowledge. The book develops a fairly complete mathematical theory of discrete Markov chains and martingales, and then in Section 4 gives some initial ideas about continuous processes. Along the way, it discusses a number of interesting applications, including gambler’s ruin, random walks on graphs, sequence waiting times, stock option pricing, branching processes, Markov chain Monte Carlo (MCMC) algorithms, and more. The book arose from point-form lecture notes. Although it was later expanded into complete sentences and paragraphs, it retains its original brevity, with short paragraphs and each “point” on its own line, to communicate ideas one at a time. Some excessively technical material is written in smaller font and may be ignored on first reading. There are also links to some animated simulations; for additional links and updates and information, visit: www.probability.ca/spbook The target audience for this book is upper-year undergraduate and graduate-level students in Statistics, Mathematics, Computer Science, Economics, Finance, Engineering, Physics, and other subjects which involve logical reasoning and mathematical foundations, and which require working knowledge of how probabilities progress in time. The prerequisites to reading this book are a solid background in basic undergraduate-level mathematics, including advanced calculus and basic linear algebra and basic real analysis (not including measure theory), plus undergraduate-level probability theory including expected values, distributions, limit theorems, etc. Appendix A contains many basic facts from elementary probability and mathematics, as needed. It is referred to frequently in the text, to clarify arguments and to fill in any knowledge gaps. Appendix B lists references for further reading about stochastic processes. Various problems are sprinkled throughout the book, and should be attempted for greater understanding. More involved problems are marked with (*). Problems marked [sol] have full solutions in Appendix C. To ease understanding and memory, I have provided colorful names for many of the results, like the Sum Lemma and Recurrence Equivalences Theorem and Closed Subset Note and Vanishing Probabilities Proposition. Hopefully the names are helpful – otherwise just use their numbers instead. But be warned that if you use these names in conversation, then readers of other books might not know what you are talking about! Acknowledgements: This text grew out of my lecturing the course STA 447/2006: Stochastic Processes at the University of Toronto over a period of many years. I thank my colleagues for giving me that opportunity. I also thank the many students who have studied these topics with me; their reactions and questions have been a major source of inspiration. Jeffrey S. Rosenthal Toronto, Canada, 2019 [email protected] www.probability.ca About the Author Jeffrey S. Rosenthal is a Professor of Statistics at the University of Toronto, specialising in Markov chain Monte Carlo (MCMC) algorithms. He received his BSc from the University of Toronto at age 20, and his PhD in Mathematics from Harvard University at age 24. He was awarded the 2006 CRM-SSC Prize, the 2007 COPSS Presidents’ Award, the 2013 SSC Gold Medal, and teaching awards at both Harvard and Toronto. He is a fellow of the Institute of Mathematical Statistics and of the Royal Society of Canada. He has published over one hundred research papers and four previous books, including Struck by Lightning for the general public which appeared in sixteen editions and ten languages and was a bestseller in Canada, and A First Look at Rigorous Probability Theory which presents probability’s measure-theoretic mathematical foundations. His website is www.probability.ca, and on Twitter he is @ProbabilityProf. (Author photo by Henry Chan.)
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