Complex-Valued Modeling in Economics and Finance 2013(经济学中的复变函数)

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俄罗斯人"怪物"写的复变函数在经济中的应用
出版日期: 2013年1月31日
First comprehensive text to discuss the fundamentals of complex-valued modeling in economics and finance Presents a systematic approach for utilizing the material in practical applications related to econometrics and financial modeling Complex-valued models are more efficient than real variables in depicting key economic processes Complex-Valued Modeling in Economics and Financeoutlines the theory, methodology, and techniques behind modeling economic processes using complex variables theory.  The theory of complex variables functions is widely used in many scientific fields, since work with complex variables can appropriately describe different complex real-life processes.  Many economic indicators and factors reflecting the properties of the same object can be represented in the form of complex variables. By describing the relationship between various indicators using the functions of these variables, new economic and financial models can be created which are often more accurate than the models of real variables.  This book pays critical attention to complex variables production in stock market modeling, modeling illegal economy, time series forecasting, complex auto-aggressive models, and economic dynamics modeling.  Very little has been published on this topic and its applications within the fields of economics and finance, and this volume appeals to graduate-level students studying economics, academic researchers in economics and finance, and economists
1 Theoretical Basis of Complex Economy . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Complex Economies as a New Branch of Economics
and Mathematical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Basic Concepts of the TFCV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Axiomatic Core of the Theory of the Complex Economy . . . . . . . 14
1.4 Basic Model of a Complex Economy . . . . . . . . . . . . . . . . . . . . . 16
1.5 Some Data on Minkowsky’s Geometry . . . . . . . . . . . . . . . . . . . . 21
1.6 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Properties of Complex Numbers of a Real Argument
and Real Functions of a Complex Argument . . . . . . . . . . . . . . . . . . 27
2.1 General Problem of Conformal Mapping
in Complex-Valued Economics . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Complex Functions of a Real Argument . . . . . . . . . . . . . . . . . . . 28
2.3 Functions of a Complex Argument: Linear Function . . . . . . . . . . 42
2.4 Power Function of a Complex Argument
with a Real Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5 Exponential Function of Complex Argument
with Imaginary Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6 Power Function of Complex Argument
with Complex Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.7 Exponential Function of a Complex Argument . . . . . . . . . . . . . . . 55
2.8 Logarithmic Function of a Complex Argument . . . . . . . . . . . . . . 59
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Conformal Mappings of Functions of a Complex Variable . . . . . . . . 63
3.1 Power Functions of a Complex Variable . . . . . . . . . . . . . . . . . . . 63
3.2 Exponential Functions of Complex Variables . . . . . . . . . . . . . . . . 75
3.3 Logarithmic Functions of Complex Variables . . . . . . . . . . . . . . . 78
3.4 Zhukovsky’s Function and Trigonometric
Complex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4 Principles of Complex-Valued Econometrics . . . . . . . . . . . . . . . . . . 87
4.1 Statistics of Random Complex Value:
Standard Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 Method of Least Squares of Complex Variables:
Standard Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.3 Correlation Analysis of Complex Variables:
Contradictions of the Standard Approach . . . . . . . . . . . . . . . . . . 95
4.4 Consistent Axioms of the Theory of Mathematical
Statistics of Random Complex Variables . . . . . . . . . . . . . . . . . . 101
4.5 Least-Squares Method from the Point of View
of the New Axiomatic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.6 Complex Pair Correlation Coefficient . . . . . . . . . . . . . . . . . . . . 112
4.7 Interpretation of Values of Complex Pair
Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.8 Assessments of Parameters of Nonlinear Econometric
Models of Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.9 Assessment of Confidence Limits of Selected Values
of Complex-Valued Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.10 Balancing Factor in Evaluating the Adequacy
of Econometric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5 Production Functions of Complex Argument . . . . . . . . . . . . . . . . . . 143
5.1 Fundamentals of Production Functions
of a Complex Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.2 Linear Complex-Valued Model of a Complex
Argument and Multicollinearity . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.3 Linear Production Function of a Complex Argument . . . . . . . . . 155
5.4 Power Production Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.5 Exponential Production Function of Complex Argument . . . . . . . 172
5.6 Logarithmic Production Function of Complex Argument . . . . . . 175
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6 Production Functions of Complex Variables . . . . . . . . . . . . . . . . . . . 181
6.1 General Provisions of the Theory of Production
Functions with Complex Variables . . . . . . . . . . . . . . . . . . . . . . . 181
6.2 Linear Production Function of Complex Variables . . . . . . . . . . . . 185
6.3 Model of Power Production Function of Complex
Variables with Real Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 194
6.4 Power Production Complex-Valued Functions
with Real Coefficients of the Diatom Plant
and Russian Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.5 Coefficients of Elasticity of the Complex Exponential
Production Function with Real Coefficients . . . . . . . . . . . . . . . . . 207
6.6 Power Production Function of Complex Variables
with Complex Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
6.7 Logarithmic Production Function of Complex Variables . . . . . . . . 221
6.8 Exponential Production Function of Complex Variables . . . . . . . . 226
6.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
7 Multifactor Complex-Valued Models of Economy . . . . . . . . . . . . . . 233
7.1 General Provisions of Complex-Valued Model Classification . . . . 233
7.2 Linear Classification Production Function . . . . . . . . . . . . . . . . . . 237
7.3 Classification Production Function of Cobb-Douglas Type . . . . . . 242
7.4 Elasticity and Other Characteristics of a Classification
Production Complex-Valued Function . . . . . . . . . . . . . . . . . . . . . 246
7.5 Classification Power Production Function . . . . . . . . . . . . . . . . . . 256
7.6 The Shadow Economy and Its Modeling by Means
of Complex-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
7.7 Formation of Complex, Multifactor Models
of Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
8 Modeling Economic Conditions of the Stock Market . . . . . . . . . . . . 269
8.1 Stock Market Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
8.2 Phase Plane and K-patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
8.3 Mathematical Models of K-Patterns . . . . . . . . . . . . . . . . . . . . . . . 287
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
9 Modeling and Forecasting of Economic Dynamics
by Complex-Valued Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
9.1 Ivan Svetunkov’s Model for Short-Term Forecasting . . . . . . . . . . 291
9.2 Complex-Valued Autoregression Models . . . . . . . . . . . . . . . . . . . 296
9.3 Solow’s Model of Economic Dynamics and Its
Complex-Valued Analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
9.4 Modeling Regional Socioeconomic Development . . . . . . . . . . . . . 304
9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

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