Essentials of Stochastic Processes

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Durrett R. -Essentials of Stochastic Processes-Springer (2012)
Richard durrett Essentials of stochastic Processes Second edition S ringer Richard durrett Duke University Department of mathematics Box90320 Durham North carolina USA ISSN1431-875X ISBN978-1-4614-3614-0 ISBN978-1-46143615-7( eBook) DOI10.1007/978-1-4614-3615-7 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012937472 o Springer Science+Business Media, LLC 1999, 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein Printed on acid-free paper SpringerispartofSpringerScience+businessMedia(www.springer.com) 70 10-Sep k10-Jun r10-May 50 at expiry 30 20 10 500520540560580600620640660680700 Preface Between the first undergraduate course in probability and the first graduate course that uses measure theory, there are a number of courses that teach Stochastic Processes to students with many different interests and with varying degrees of mathematical sophistication. To allow readers(and instructors) to choose their own level of detail, many of the proofs begin with a nonrigorous answer to the question Why is this true? followed by a Proof that fills in the missing details. As it is possible to drive a car without knowing about the working of the internal combustion engine, it is also possible to apply the theory of markov chains without knowing the details of the proofs. It is my personal philosophy that probability theory was developed to solve problems, so most of our effort will be spent on analyzing examples. Readers who want to master the subject will have to do more than a few of the 20 dozen carefully chosen exercises This book began as notes I typed in the spring of 1997 as I was teaching ORiE 361 at Cornell for the second time. In Spring 2009, the mathematics department there introduced its own version of this course. math 474. This started me on the task of preparing the second edition. The plan was to have this finished in Spring 2010 after the second time I taught the course, but when May rolled around completing the book lost out to getting ready to move to Durham after 25 years in Ithaca. In the Fall of 2011, i taught dukes version of the course, Math 216,to 20 undergrads and 12 graduate students and over the Christmas break the second edition was completed The second edition differs substantially from the first, though curiously the length and the number of problems has remained roughly constant. Throughout the book there are many new examples and problems, with solutions that use the ti-83 to eliminate the tedious details of solving linear equations by hand. My students tell me I should just use matlab and maybe I will for the next edition The Markov chains chapter has been reorganized. The chapter on Poisson processes has moved up from third to second, and is now followed by a treatment of the closely related topic of renewal theory. Continuous time Markov chains remain fourth, with a new section on exit distributions and hitting times, and reduced coverage of queueing networks. Martingales, a difficult subject for students at this VIll Preface level, now comes fifth, in order to set the stage for their use in a new sixth chapter on mathematical finance. The treatment of finance expands the two sections of the previous treatment to include American options and the the capital asset pricin model. Brownian motion makes a cameo appearance in the discussion of the black- Scholes theorem, but in contrast to the previous edition, is not discussed in detail As usual the second edition has profited from people who have told me about typos over the last dozen years. If you find new ones, email: rtd @math. duke. edu Rick durrett Contents 1 Markov chains 1.1 Definitions and Examples............. 1.2 Multistep Transition Probabilities 8 1. 3 Classification of states 12 1. 4 Stationary distributi 20 1.5 Limit behavior 26 1.6 Special Example 34 1.6.1 Doubly stochastic Chains 34 1. 6.2 Detailed Balance Condition. ..................................37 1.6.3 Reversibility 42 1.6.4 The Metropolis-Ha Algorith 43 1. 7 Proofs of the main theorems 46 1. 8 Exit distributions 1.9 Exit Times 58 1.10 Infinite State s 64 1. 11 Chapter summary 1. 12 Exercises 75 2 Poisson Processes............................... 93 2.1 Exponential distributi 93 2.2 Defining the Poisson Process 97 2.3 Compound Poisson processes 103 2.4 Transformations 106 2.4.1 Thinning 106 2.4.2Su 107 2.4.3 Conditioning....………108 Chapter summary… 110 2.6 Exe 111 3 Renewal processes 119 3. 1 Laws of Large Numbers 119

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