Optimal Control Models in Finance A New Computational Approach
The determination of optimal financing and investment strategies (optimal capital structure or optimal mix of funds, optimal portfolio choice, etc.) for corporations and the economy are important for efficient allocation of resources in the economy. Optim
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Applied Optimization Volume 95 Series Editors: Panos M. Pardalos University of Florida, U.S.A. Donald W. Hearn University of Florida, U.S.A.
OPTIMAL CONTROL MODELS IN FINANCE A New Computational Approach by
PING CHEN Victoria University, Melbourne, Australia
SARDAR M.N. ISLAM Victoria University, Melbourne, Australia
Springer
eBook ISBN: Print ISBN:
0-387-23570-1 0-387-23569-8
©2005 Springer Science + Business Media, Inc. Print ©2005 Springer Science + Business Media, Inc. Boston All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America
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Contents
List of Figures List of Tables Preface Introduction 1. OPTIMAL CONTROL MODELS 1 An Optimal Control Model of Finance (Karush) Kuhn-Tucker Condition 2 Pontryagin Theorem 3 Bang-Bang Control 4 5 6 7 8
Singular Arc Indifference Principle Different Approaches to Optimal Control Problems Conclusion
2. THE STV APPROACH TO FINANCIAL OPTIMAL CONTROL MODELS 1 Introduction 2 Piecewise-linear Transformation Non-linear Time Scale Transformation 3 A Computer Software Package Used in this Study 4 5 An Optimal Control Problem When the Control can only Take the Value 0 or 1 6 Approaches to Bang-Bang Optimal Control with a Cost of Changing Control An Investment Planning Model and Results 7
ix xi xiii xv 1 2 4 6 7 7 8 10 20 21 21 21 23 25 26 27 30
OPTIMAL CONTROL MODELS IN FINANCE
vi
8
Financial Implications and Conclusion
3. A FINANCIAL OSCILLATOR MODEL 1 Introduction 2 Controlling a Damped Oscillator in a Financial Model Oscillator Transformation of the Financial Model 3 4 Computational Algorithm: The Steps Financial Control Pattern 5 Computing the Financial Model: Results and Analysis 6 Financial Investment Implications and Conclusion 7 4. AN OPTIMAL CORPORATE FINANCING MODEL 1 Introduction 2 Problem Description Analytical Solution 3 4 Penalty Terms Transformations for the Computer Software Package for the 5 Finance Model Computational Algorithms for the Non-linear Optimal Control 6 Problem Computing Results and Conclusion 7 Optimal Financing Implications 8 Conclusion 9
36 39 39 40 41 44 47 47 89 91 91 91 94 98 99 101 104 107 108
5. FURTHER COMPUTATIONAL EXPERIMENTS AND RESULTS 1 Introduction 2 Different Fitting Functions The Financial Oscillator Model when the Control Takes Three 3 Values 4 Conclusion
109 109 109
6. CONCLUSION
141
Appendices A CSTVA Program List 1 Program A: Investment Model in Chapter 2 2 Program B: Financial Oscillator Model in Chapter 3 Program C: Optimal Financing Model in Chapter 4 3 Program D: Three Value-Control Model in Chapter 5 4
145 145 145 149 153 156
120 139
Contents
vii
B Some Computation Results 1 Results for Program A Results for Program B 2 Results for Program C 3 Results for Program D 4 C Differential Equation
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