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Mathematical Economics: Application of Fractional Calculus

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Financial Mathematics".

Deadline for manuscript submissions: closed (20 February 2020) | Viewed by 52432

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Prof. Dr. Vasily E. Tarasov

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Guest Editor

Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, 119991 Moscow, Russia
Interests: fractional calculus; fractional dynamics; mathematical economics; quantum theory; theoretical physics; processes with memory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional calculus is a branch of mathematics that studies the properties of differential and integral operators that are characterized by real or complex orders. Methods of fractional calculus are powerful tools for describing the processes and systems with memory and nonlocality. There are various types of fractional integral and differential operators that are proposed by Riemann, Liouville, Grunwald, Letnikov, Sonine, Marchaud, Weyl, Riesz, Hadamard, Kober, Erdelyi, Caputo and other mathematicians. The fractional derivatives have a set of nonstandard properties such as a violation of the standard product and chain rules. The violation of the standard form of the product rule is a main characteristic property of derivatives of non-integer orders that allows us to describe complex properties of processes and systems.

Recently, fractional integro-differential equations are actively used to describe a wide class of economical processes with power-law memory and spatial nonlocality. Generalizations of basic economic concepts and notions the economic processes with memory were proposed. New mathematical models with continuous time are proposed to describe the economic dynamics with long memory.

The purpose of this Special Issue is to create a collection of articles reflecting the latest mathematical and conceptual developments in mathematical economics with memory and non-locality based on applications of modern fractional calculus.

Prof. Dr. Vasily E. Tarasov
Guest Editor

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Keywords

Fractional calculus
Mathematical economics
Fractional derivative
Fractional integral
Fractional difference
Long memory
Spatial non-locality
Economic growth models
Processes with memory

Published Papers (13 papers)

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Editorial

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3 pages, 158 KiB

Open AccessEditorial

Mathematical Economics: Application of Fractional Calculus

by Vasily E. Tarasov

Mathematics 2020, 8(5), 660; https://0-doi-org.brum.beds.ac.uk/10.3390/math8050660 - 27 Apr 2020

Cited by 41 | Viewed by 4017

Abstract

Mathematical economics is a theoretical and applied science in which economic objects, processes, and phenomena are described by using mathematically formalized language [...] Full article

(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)

Research

Jump to: Editorial, Review

18 pages, 3507 KiB

Open AccessArticle

Deep Assessment Methodology Using Fractional Calculus on Mathematical Modeling and Prediction of Gross Domestic Product per Capita of Countries

by Ertuğrul Karaçuha, Vasil Tabatadze, Kamil Karaçuha, Nisa Özge Önal and Esra Ergün

Mathematics 2020, 8(4), 633; https://0-doi-org.brum.beds.ac.uk/10.3390/math8040633 - 20 Apr 2020

Cited by 9 | Viewed by 3360

Abstract

In this study, a new approach for time series modeling and prediction, “deep assessment methodology,” is proposed and the performance is reported on modeling and prediction for upcoming years of Gross Domestic Product (GDP) per capita. The proposed methodology expresses a function with the finite summation of its previous values and derivatives combining fractional calculus and the Least Square Method to find unknown coefficients. The dataset of GDP per capita used in this study includes nine countries (Brazil, China, India, Italy, Japan, the UK, the USA, Spain and Turkey) and the European Union. The modeling performance of the proposed model is compared with the Polynomial model and the Fractional model and prediction performance is compared to a special type of neural network, Long Short-Term Memory (LSTM), that used for time series. Results show that using Deep Assessment Methodology yields promising modeling and prediction results for GDP per capita. The proposed method is outperforming Polynomial model and Fractional model by 1.538% and by 1.899% average error rates, respectively. We also show that Deep Assessment Method (DAM) is superior to plain LSTM on prediction for upcoming GDP per capita values by 1.21% average error. Full article

(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)

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17 pages, 961 KiB

Open AccessArticle

Fractional Dynamics and Pseudo-Phase Space of Country Economic Processes

by José A. Tenreiro Machado, Maria Eugénia Mata and António M. Lopes

Mathematics 2020, 8(1), 81; https://0-doi-org.brum.beds.ac.uk/10.3390/math8010081 - 03 Jan 2020

Cited by 9 | Viewed by 2412

Abstract

In this paper, the fractional calculus (FC) and pseudo-phase space (PPS) techniques are combined for modeling the dynamics of world economies, leading to a new approach for forecasting a country’s gross domestic product. In most market economies, the decline of the post-war prosperity brought challenging rivalries to the Western world. Considerable social, political, and military unrest is today spreading in major capital cities of the world. As global troubles including mass migrations and more abound, countries’ performance as told by PPS approaches can help to assess national ambitions, commercial aggression, or hegemony in the current global environment. The 1973 oil shock was the turning point for a long-run crisis. A PPS approach to the last five decades (1970–2018) demonstrates that convergence has been the rule. In a sample of 15 countries, Turkey, Russia, Mexico, Brazil, Korea, and South Africa are catching-up to the US, Canada, Japan, Australia, Germany, UK, and France, showing similarity in many respects with these most developed countries. A substitution of the US role as great power in favor of China may still be avoided in the next decades, while India remains in the tail. The embedding of the two mathematical techniques allows a deeper understanding of the fractional dynamics exhibited by the world economies. Additionally, as a byproduct we obtain a foreseeing technique for estimating the future evolution based on the memory of the time series. Full article

(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)

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21 pages, 1057 KiB

Open AccessEditor’s ChoiceArticle

Fractional Derivatives for Economic Growth Modelling of the Group of Twenty: Application to Prediction

by Inés Tejado, Emiliano Pérez and Duarte Valério

Mathematics 2020, 8(1), 50; https://0-doi-org.brum.beds.ac.uk/10.3390/math8010050 - 01 Jan 2020

Cited by 17 | Viewed by 3604

Abstract

This paper studies the economic growth of the countries in the Group of Twenty (G20) in the period 1970–2018. It presents dynamic models for the world’s most important national economies, including for the first time several economies which are not highly developed. Additional care has been devoted to the number of years needed for an accurate short-term prediction of future outputs. Integer order and fractional order differential equation models were obtained from the data. Their output is the gross domestic product (GDP) of a G20 country. Models are multi-input; GDP is found from all or some of the following variables: country’s land area, arable land, population, school attendance, gross capital formation (GCF), exports of goods and services, general government final consumption expenditure (GGFCE), and broad money (M3). Results confirm the better performance of fractional models. This has been established employing several summary statistics. Fractional models do not require increasing the number of parameters, neither do they sacrifice the ability to predict GDP evolution in the short-term. It was found that data over 15 years allows building a model with a satisfactory prediction of the evolution of the GDP. Full article

(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)

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57 pages, 1125 KiB

Open AccessFeature PaperArticle

Econophysics and Fractional Calculus: Einstein’s Evolution Equation, the Fractal Market Hypothesis, Trend Analysis and Future Price Prediction

by Jonathan Blackledge, Derek Kearney, Marc Lamphiere, Raja Rani and Paddy Walsh

Mathematics 2019, 7(11), 1057; https://0-doi-org.brum.beds.ac.uk/10.3390/math7111057 - 04 Nov 2019

Cited by 12 | Viewed by 3232

Abstract

This paper examines a range of results that can be derived from Einstein’s evolution equation focusing on the effect of introducing a Lévy distribution into the evolution equation. In this context, we examine the derivation (derived exclusively from the evolution equation) of the classical and fractional diffusion equations, the classical and generalised Kolmogorov–Feller equations, the evolution of self-affine stochastic fields through the fractional diffusion equation, the fractional Poisson equation (for the time independent case), and, a derivation of the Lyapunov exponent and volatility. In this way, we provide a collection of results (which includes the derivation of certain fractional partial differential equations) that are fundamental to the stochastic modelling associated with elastic scattering problems obtained under a unifying theme, i.e., Einstein’s evolution equation. This includes an analysis of stochastic fields governed by a symmetric (zero-mean) Gaussian distribution, a Lévy distribution characterised by the Lévy index

γ \in [0, 2]

and the derivation of two impulse response functions for each case. The relationship between non-Gaussian distributions and fractional calculus is examined and applications to financial forecasting under the fractal market hypothesis considered, the reader being provided with example software functions (written in MATLAB) so that the results presented may be reproduced and/or further investigated. Full article

(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)

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10 pages, 910 KiB

Open AccessArticle

Stability and Bifurcation of a Delayed Time-Fractional Order Business Cycle Model with a General Liquidity Preference Function and Investment Function

by Yingkang Xie, Zhen Wang and Bo Meng

Mathematics 2019, 7(9), 846; https://0-doi-org.brum.beds.ac.uk/10.3390/math7090846 - 13 Sep 2019

Cited by 13 | Viewed by 2855

Abstract

In this paper, the business cycle (BC) is described by a delayed time-fractional-order model (DTFOM) with a general liquidity preference function and an investment function. Firstly, the existence and uniqueness of the DTFOM solution are proven. Then, some conditions are presented to guarantee that the positive equilibrium point of DTFOM is locally stable. In addition, Hopf bifurcation is obtained by a new method, where the time delay is regarded as the bifurcation parameter. Finally, a numerical example of DTFOM is given to verify the effectiveness of the proposed model and methods. Full article

(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)

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23 pages, 479 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Applications of the Fractional Diffusion Equation to Option Pricing and Risk Calculations

by Jean-Philippe Aguilar, Jan Korbel and Yuri Luchko

Mathematics 2019, 7(9), 796; https://0-doi-org.brum.beds.ac.uk/10.3390/math7090796 - 01 Sep 2019

Cited by 20 | Viewed by 5084

Abstract

In this article, we first provide a survey of the exponential option pricing models and show that in the framework of the risk-neutral approach, they are governed by the space-fractional diffusion equation. Then, we introduce a more general class of models based on the space-time-fractional diffusion equation and recall some recent results in this field concerning the European option pricing and the risk-neutral parameter. We proceed with an extension of these results to the class of exotic options. In particular, we show that the call and put prices can be expressed in the form of simple power series in terms of the log-forward moneyness and the risk-neutral parameter. Finally, we provide the closed-form formulas for the first and second order risk sensitivities and study the dependencies of the portfolio hedging and profit-and-loss calculations upon the model parameters. Full article

(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)

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6 pages, 255 KiB

Open AccessEditor’s ChoiceArticle

The Application of Fractional Calculus in Chinese Economic Growth Models

by Hao Ming, JinRong Wang and Michal Fečkan

Mathematics 2019, 7(8), 665; https://0-doi-org.brum.beds.ac.uk/10.3390/math7080665 - 25 Jul 2019

Cited by 28 | Viewed by 4119

Abstract

In this paper, we apply Caputo-type fractional order calculus to simulate China’s gross domestic product (GDP) growth based on R software, which is a free software environment for statistical computing and graphics. Moreover, we compare the results for the fractional model with the integer order model. In addition, we show the importance of variables according to the BIC criterion. The study shows that Caputo fractional order calculus can produce a better model and perform more accurately in predicting the GDP values from 2012–2016. Full article

(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)

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8 pages, 224 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Growth Equation of the General Fractional Calculus

by Anatoly N. Kochubei and Yuri Kondratiev

Mathematics 2019, 7(7), 615; https://0-doi-org.brum.beds.ac.uk/10.3390/math7070615 - 11 Jul 2019

Cited by 41 | Viewed by 2821

Abstract

We consider the Cauchy problem

(D_{(k)} u) (t) = λ u (t)

u (0) = 1

, where

D_{(k)}

is the general convolutional derivative introduced in the paper [...] Read more.

We consider the Cauchy problem

(D_{(k)} u) (t) = λ u (t)

u (0) = 1

, where

D_{(k)}

is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory 71 (2011), 583–600),

λ > 0

. The solution is a generalization of the function

t \mapsto E_{α} (λ t^{α})

, where

0 < α < 1

E_{α}

is the Mittag–Leffler function. The asymptotics of this solution, as

t \to \infty

, are studied. Full article

(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)

11 pages, 478 KiB

Open AccessFeature PaperArticle

The Mittag-Leffler Fitting of the Phillips Curve

by Tomas Skovranek

Mathematics 2019, 7(7), 589; https://0-doi-org.brum.beds.ac.uk/10.3390/math7070589 - 01 Jul 2019

Cited by 8 | Viewed by 2926

Abstract

In this paper, a mathematical model based on the one-parameter Mittag-Leffler function is proposed to be used for the first time to describe the relation between the unemployment rate and the inflation rate, also known as the Phillips curve. The Phillips curve is in the literature often represented by an exponential-like shape. On the other hand, Phillips in his fundamental paper used a power function in the model definition. Considering that the ordinary as well as generalised Mittag-Leffler function behave between a purely exponential function and a power function it is natural to implement it in the definition of the model used to describe the relation between the data representing the Phillips curve. For the modelling purposes the data of two different European economies, France and Switzerland, were used and an “out-of-sample” forecast was done to compare the performance of the Mittag-Leffler model to the performance of the power-type and exponential-type model. The results demonstrate that the ability of the Mittag-Leffler function to fit data that manifest signs of stretched exponentials, oscillations or even damped oscillations can be of use when describing economic relations and phenomenons, such as the Phillips curve. Full article

(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)

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Review

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9 pages, 413 KiB

Open AccessReview

On the Advent of Fractional Calculus in Econophysics via Continuous-Time Random Walk

by Francesco Mainardi

Mathematics 2020, 8(4), 641; https://0-doi-org.brum.beds.ac.uk/10.3390/math8040641 - 21 Apr 2020

Cited by 15 | Viewed by 2192

Abstract

In this survey article, at first, the author describes how he was involved in the late 1990s on Econophysics, considered in those times an emerging science. Inside a group of colleagues the methods of the Fractional Calculus were developed to deal with the continuous-time random walks adopted to model the tick-by-tick dynamics of financial markets Then, the analytical results of this approach are presented pointing out the relevance of the Mittag-Leffler function. The consistence of the theoretical analysis is validated with fitting the survival probability for certain futures (BUND and BTP) traded in 1997 at LIFFE, London. Most of the theoretical and numerical results (including figures) reported in this paper were presented by the author at the first Nikkei symposium on Econophysics, held in Tokyo on November 2000 under the title “Empirical Science of Financial Fluctuations” on behalf of his colleagues and published by Springer. The author acknowledges Springer for the license permission of re-using this material. Full article

(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)

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50 pages, 2766 KiB

Open AccessFeature PaperEditor’s ChoiceReview

Rules for Fractional-Dynamic Generalizations: Difficulties of Constructing Fractional Dynamic Models

by Vasily E. Tarasov

Mathematics 2019, 7(6), 554; https://0-doi-org.brum.beds.ac.uk/10.3390/math7060554 - 18 Jun 2019

Cited by 30 | Viewed by 3999

Abstract

This article is a review of problems and difficulties arising in the construction of fractional-dynamic analogs of standard models by using fractional calculus. These fractional generalizations allow us to take into account the effects of memory and non-locality, distributed lag, and scaling. We formulate rules (principles) for constructing fractional generalizations of standard models, which were described by differential equations of integer order. Important requirements to building fractional generalization of dynamical models (the rules for “fractional-dynamic generalizers”) are represented as the derivability principle, the multiplicity principle, the solvability and correspondence principles, and the interpretability principle. The characteristic properties of fractional derivatives of non-integer order are the violation of standard rules and properties that are fulfilled for derivatives of integer order. These non-standard mathematical properties allow us to describe non-standard processes and phenomena associated with non-locality and memory. However, these non-standard properties lead to restrictions in the sequential and self-consistent construction of fractional generalizations of standard models. In this article, we give examples of problems arising due to the non-standard properties of fractional derivatives in construction of fractional generalizations of standard dynamic models in economics. Full article

(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)

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28 pages, 356 KiB

Open AccessFeature PaperEditor’s ChoiceReview

On History of Mathematical Economics: Application of Fractional Calculus

by Vasily E. Tarasov

Mathematics 2019, 7(6), 509; https://0-doi-org.brum.beds.ac.uk/10.3390/math7060509 - 04 Jun 2019

Cited by 136 | Viewed by 9352

Abstract

Modern economics was born in the Marginal revolution and the Keynesian revolution. These revolutions led to the emergence of fundamental concepts and methods in economic theory, which allow the use of differential and integral calculus to describe economic phenomena, effects, and processes. At the present moment the new revolution, which can be called “Memory revolution”, is actually taking place in modern economics. This revolution is intended to “cure amnesia” of modern economic theory, which is caused by the use of differential and integral operators of integer orders. In economics, the description of economic processes should take into account that the behavior of economic agents may depend on the history of previous changes in economy. The main mathematical tool designed to “cure amnesia” in economics is fractional calculus that is a theory of integrals, derivatives, sums, and differences of non-integer orders. This paper contains a brief review of the history of applications of fractional calculus in modern mathematical economics and economic theory. The first stage of the Memory Revolution in economics is associated with the works published in 1966 and 1980 by Clive W. J. Granger, who received the Nobel Memorial Prize in Economic Sciences in 2003. We divide the history of the application of fractional calculus in economics into the following five stages of development (approaches): ARFIMA; fractional Brownian motion; econophysics; deterministic chaos; mathematical economics. The modern stage (mathematical economics) of the Memory revolution is intended to include in the modern economic theory new economic concepts and notions that allow us to take into account the presence of memory in economic processes. The current stage actually absorbs the Granger approach based on ARFIMA models that used only the Granger–Joyeux–Hosking fractional differencing and integrating, which really are the well-known Grunwald–Letnikov fractional differences. The modern stage can also absorb other approaches by formulation of new economic notions, concepts, effects, phenomena, and principles. Some comments on possible future directions for development of the fractional mathematical economics are proposed. Full article

(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)

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