Lecture Notes in Control and Information Sciences 467 Selim S. Hacısalihzade Control Engineering and Finance Lecture Notes in Control and Information Sciences Volume 467 Series editors Frank Allgöwer, Stuttgart, Germany Manfred Morari, Zürich, Switzerland Series Advisory Boards P. Fleming, University of Sheffield, UK P. Kokotovic, University of California, Santa Barbara, CA, USA A.B. Kurzhanski, Moscow State University, Russia H. Kwakernaak, University of Twente, Enschede, The Netherlands A. Rantzer, Lund Institute of Technology, Sweden J.N. Tsitsiklis, MIT, Cambridge, MA, USA About this Series This series aims to report new developments in the fields of control and information sciences—quickly, informally and at a high level. The type of material considered for publication includes: 1. Preliminary drafts of monographs and advanced textbooks 2. Lectures on a new field, or presenting a new angle on a classical field 3. Research reports 4. Reports of meetings, provided they are (a) of exceptional interest and (b) devoted to a specific topic. The timeliness of subject material is very important. More information about this series at http://www.springer.com/series/642 Selim S. Hacısalihzade Control Engineering and Finance 123 Selim S. Hacısalihzade Department of Electrical and Electronics Engineering Boğaziçi University Bebek, Istanbul Turkey ISSN 0170-8643 ISSN 1610-7411 (electronic) Lecture Notes in Control and Information Sciences ISBN 978-3-319-64491-2 ISBN 978-3-319-64492-9 (eBook) https://doi.org/10.1007/978-3-319-64492-9 Library of Congress Control Number: 2017949161 © Springer International Publishing AG 2018 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. 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Acknowledgements There is a long list of people to acknowledge and thank for their support in preparing this book. I want to begin by thanking Jürg Tödtli who supported my interest in the field of quantitative finance—even though I did not know the term at that time—while we were at the Institute of Automatic Control at the ETH Zurich many decades ago. This interest was triggered and then re-triggered in countless discussions over the years with my uncle ErgünYüksel. Special thanks are certainly due to Manfred Morari, the former head of the Institute of Automatic Control at the ETH Zurich, who encouraged me to write this book and to publish it in the Lecture Notes in Control and Information Science series of Springer Verlag and who also offered me infrastructure and library access during the preparation of the manuscript. Very special thanks go to Florian Herzog of Swissquant who supported me while I was writing this book by offering the use of his lecture notes of Stochastic Control, a graduate course he held at the ETH Zurich. I did so with gratitude in Chapters 5 and 9. The data for the empirical study reported in Chapter 8 were graciously supplied by Özgür Tanrverdi of Access Turkey Opportunities Fund for several years, to whom I am indebted. I also want to thank Jens Galschiøt, the famous Danish sculptor for allowing me to use a photograph of his impressive and inspiring sculpture “Survival of the Fattest” to illustrate the inequality in global wealth distribution. Parts of this book evolved from a graduate class I gave at Boğaziçi University in Istanbul during the last years and from project work by many students there, notably Efe Doğan Yılmaz, Ufuk Uyan, Ceren Sevinç, Mehmet Hilmi Elihoş, Yusuf Koçyiğit and Yasin Çotur. I am most grateful to Yasin, a Ph.D. candidate at Imperial College in London now, who helped with calculations and with valuable feedback on earlier versions of the manuscript. I am indebted to my former student Yaşar Baytın, my old friend Sedat Ölçer, Head of Computer Science and Engineering at Bilgi University in Istanbul, Bülent Sankur, Professor Emeritus of Electrical and Electronics Engineering at Boğaziçi vii viii Acknowledgements University, and especially my dear wife Hande Hacısalihzade for proofreading parts of the manuscript and their most valuable suggestions. I am grateful to Petra Jantzen and Shahid S. Mohammed at Springer for their assistance with the printing of this volume. Hande, of course, also deserves special thanks for inspiring me (and certainly not only for writing limericks!), hours of lively discussions, her encouragement, and her endless support during the preparation of this book. Selim S. Hacısalihzade Istanbul 2017 Contents 1 Introduction.............................................. 1 1.1 Control Engineering and Finance ......................... 1 1.2 Outline ............................................. 3 2 Modeling and Identification ................................. 7 2.1 Introduction ......................................... 7 2.2 What Is a Model? ..................................... 8 2.3 Modeling Process ..................................... 17 2.3.1 Stock Prices................................... 19 2.3.2 Lessons Learned ............................... 22 2.4 Parameter Identification ................................ 23 2.5 Mathematics of Parameter Identification .................... 24 2.5.1 Basics of Extremes ............................. 25 2.5.2 Optimization with Constraints ..................... 28 2.6 Numerical Methods for Parameter Identification.............. 31 2.6.1 Golden Section ................................ 32 2.6.2 Successive Parameter Optimization ................. 32 2.7 Model Validation ..................................... 34 2.8 Summary ........................................... 34 2.9 Exercises............................................ 36 3 Probability and Stochastic Processes .......................... 39 3.1 Introduction ......................................... 39 3.2 History and Kolmogorov’s Axioms ....................... 40 3.3 Random Variables and Probability Distributions.............. 41 3.3.1 Random Variables .............................. 41 3.3.2 Probability Distribution of a Discrete Random Variable...................................... 43 3.3.3 Binomial Distribution ........................... 43 3.3.4 Distribution Functions ........................... 45 ix x Contents 3.3.5 Multidimensional Distribution Functions and Independence .............................. 47 3.3.6 Expected Value and Further Moments............... 47 3.3.7 Correlation.................................... 51 3.3.8 Normal Distribution............................. 53 3.3.9 Central Limit Theorem .......................... 53 3.3.10 Log-Normal Distribution ......................... 57 3.4 Stochastic Processes ................................... 58 3.5 Mathematical Description of Stochastic Processes with Distribution Functions ............................. 60 3.6 Stationary and Ergodic Processes ......................... 65 3.7 Spectral Density ...................................... 67 3.8 Some Special Processes ................................ 68 3.8.1 Normal (Gaussian) Process ....................... 68 3.8.2 Markov Process................................ 69 3.8.3 Process with Independent Increments ............... 72 3.8.4 Wiener Process ................................ 73 3.8.5 Gaussian White Noise ........................... 74 3.9 Analysis of Stochastic Processes.......................... 74 3.9.1 Convergence .................................. 74 3.9.2 Continuity .................................... 75 3.9.3 Differentiability ................................ 75 3.9.4 Integrability ................................... 75 3.9.5 Brief Summary ................................ 76 3.10 Exercises............................................ 76 4 Optimal Control .......................................... 83 4.1 Introduction ......................................... 83 4.2 Calculus of Variations ................................. 85 4.2.1 Subject Matter ................................. 85 4.2.2 Fixed Endpoint Problem ......................... 86 4.2.3 Variable Endpoint Problem ....................... 88 4.2.4 Variation Problem with Constraints ................. 90 4.3 Optimal Dynamic Systems .............................. 91 4.3.1 Fixed Endpoint Problem ......................... 91 4.3.2 Variable Endpoint Problem ....................... 95 4.3.3