ix
List of Figures
List of Tables
xi
Preface
xiii
Introduction xv
1. OPTIMAL CONTROL MODELS 1
2
1 An Optimal Control Model of Finance
4
2 (Karush) Kuhn-Tucker Condition
Pontryagin Theorem
6
3
Bang-Bang Control
4 7
7
Singular Arc
5
6 8
IndifferencePrinciple
Different Approaches to Optimal Control Problems
10
7
8 Conclusion 20
2. THE STV APPROACH TO FINANCIAL OPTIMAL CONTROL
MODELS
21
1
Introduction
21
2 Piecewise-linear Transformation
21
Non-linear Time Scale Transformation
3 23
A Computer Software Package Used in this Study
4
25
5
An Optimal Control Problem When the Control can only Take
the Value 0 or 1
26
6 Approaches to Bang-Bang OptimalControl with a Cost of
Changing Control
27
An Investment Planning Model and Results
7
30
vi OPTIMAL CONTROL MODELS IN FINANCE
8 Financial Implications and Conclusion
36
3. A FINANCIAL OSCILLATOR MODEL
39
1
Introduction
39
2 Controlling a Damped Oscillator in a Financial Model
40
Oscillator Transformation of the Financial Model
3 41
4
Computational Algorithm: The Steps
44
Financial Control Pattern
5
47
Computing the Financial Model: Results and Analysis
6 47
Financial Investment Implications and Conclusion
7
89
4. AN OPTIMAL CORPORATE FINANCING MODEL
91
1
Introduction 91
2 Problem Description
91
Analytical Solution
3 94
4
Penalty Terms
98
Transformations for the Computer Software Package for the
5
Finance Model
99
Computational Algorithms for the Non-linear Optimal Control
6
Problem
101
Computing Results and Conclusion
104
7
Optimal Financing Implications
8 107
Conclusion
9 108
109
5. FURTHER COMPUTATIONAL EXPERIMENTS AND RESULTS
1 Introduction
109
2
Different Fitting Functions 109
The Financial Oscillator Model when the Control Takes Three
3
Values 120
4 Conclusion 139
6. CONCLUSION 141
145
Appendices
145
A CSTVA Program List
1
Program A: Investment Model in Chapter 2 145
2 Program B: Financial Oscillator Model in Chapter 3 149
Program C: Optimal Financing Model in Chapter 4 153
3
156
Program D: Three Value-Control Model in Chapter 5
4
Contents vii
161
B Some Computation Results
1 Results for Program A 161
Results for Program B
2 163
167
Results for Program C
3
Results for Program D
4 175
Differential Equation Solver from the SCOM Package 181
C
183
SCOM Package
D
Format of Problem Optimization
E 189
191
A Sample Test Problem
F
References 193
Index
199
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