IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v7y2019i6p509-d237116.html
   My bibliography  Save this article

On History of Mathematical Economics: Application of Fractional Calculus

Author

Listed:
  • Vasily E. Tarasov

    (Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia)

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.

Suggested Citation

  • Vasily E. Tarasov, 2019. "On History of Mathematical Economics: Application of Fractional Calculus," Mathematics, MDPI, vol. 7(6), pages 1-28, June.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:6:p:509-:d:237116
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/7/6/509/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/7/6/509/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Enrico Scalas & Rudolf Gorenflo & Hugh Luckock & Francesco Mainardi & Maurizio Mantelli & Marco Raberto, 2004. "Anomalous waiting times in high-frequency financial data," Quantitative Finance, Taylor & Francis Journals, vol. 4(6), pages 695-702.
    2. Scalas, Enrico, 2006. "The application of continuous-time random walks in finance and economics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 362(2), pages 225-239.
    3. Caputo Michele, 2012. "The Convergence of Economic Developments," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 16(2), pages 1-23, April.
    4. Tarasov, Vasily E. & Tarasova, Valentina V., 2018. "Macroeconomic models with long dynamic memory: Fractional calculus approach," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 466-486.
    5. Robert M. Solow, 1956. "A Contribution to the Theory of Economic Growth," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 70(1), pages 65-94.
    6. Hameed Ur Rehman & Maslina Darus & Jamal Salah, 2018. "A Note on Caputo’s Derivative Operator Interpretation in Economy," Journal of Applied Mathematics, Hindawi, vol. 2018, pages 1-7, October.
    7. Scalas, Enrico & Gorenflo, Rudolf & Mainardi, Francesco, 2000. "Fractional calculus and continuous-time finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 376-384.
    8. J. A. Tenreiro Machado & Alexandra M. S. F. Galhano & Juan J. Trujillo, 2014. "On development of fractional calculus during the last fifty years," Scientometrics, Springer;Akadémiai Kiadó, vol. 98(1), pages 577-582, January.
    9. Valentina V. Tarasova & Vasily E. Tarasov, 2017. "Risk Aversion for Investors with Memory: Hereditary Generalizations of Arrow-Pratt Measure," Finansovyj žhurnal — Financial Journal, Financial Research Institute, Moscow 125375, Russia, issue 2, pages 46-63, April.
    10. Valentina V. Tarasova & Vasily E. Tarasov, 2017. "Accelerators in macroeconomics: Comparison of discrete and continuous approaches," Papers 1712.09605, arXiv.org.
    11. Vasily E. Tarasov & Valentina V. Tarasova, 2016. "Long and Short Memory in Economics: Fractional-Order Difference and Differentiation," Papers 1612.07903, arXiv.org, revised Aug 2017.
    12. Jajarmi, Amin & Hajipour, Mojtaba & Baleanu, Dumitru, 2017. "New aspects of the adaptive synchronization and hyperchaos suppression of a financial model," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 285-296.
    13. Muniandy, S.V. & Lim, S.C. & Murugan, R., 2001. "Inhomogeneous scaling behaviors in Malaysian foreign currency exchange rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 301(1), pages 407-428.
    14. Laskin, Nick, 2000. "Fractional market dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 482-492.
    15. Banerjee, Anindya & Urga, Giovanni, 2005. "Modelling structural breaks, long memory and stock market volatility: an overview," Journal of Econometrics, Elsevier, vol. 129(1-2), pages 1-34.
    16. Klüppelberg, Claudia & Kühn, Christoph, 2004. "Fractional Brownian motion as a weak limit of Poisson shot noise processes--with applications to finance," Stochastic Processes and their Applications, Elsevier, vol. 113(2), pages 333-351, October.
    17. T. W. Swan, 1956. "ECONOMIC GROWTH and CAPITAL ACCUMULATION," The Economic Record, The Economic Society of Australia, vol. 32(2), pages 334-361, November.
    18. Valentina V. Tarasova & Vasily E. Tarasov, 2017. "Economic interpretation of fractional derivatives," Papers 1712.09575, arXiv.org.
    19. Raberto, Marco & Scalas, Enrico & Mainardi, Francesco, 2002. "Waiting-times and returns in high-frequency financial data: an empirical study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 314(1), pages 749-755.
    20. Tarasova, Valentina V. & Tarasov, Vasily E., 2017. "Logistic map with memory from economic model," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 84-91.
    21. Enrico Scalas, 2006. "Five Years of Continuous-time Random Walks in Econophysics," Lecture Notes in Economics and Mathematical Systems, in: Akira Namatame & Taisei Kaizouji & Yuuji Aruka (ed.), The Complex Networks of Economic Interactions, pages 3-16, Springer.
    22. Chen, Wei-Ching, 2008. "Dynamics and control of a financial system with time-delayed feedbacks," Chaos, Solitons & Fractals, Elsevier, vol. 37(4), pages 1198-1207.
    23. Shichang Ma & Yufeng Xu & Wei Yue, 2012. "Numerical Solutions of a Variable-Order Fractional Financial System," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-14, September.
    24. Chen, Wei-Ching, 2008. "Nonlinear dynamics and chaos in a fractional-order financial system," Chaos, Solitons & Fractals, Elsevier, vol. 36(5), pages 1305-1314.
    25. Liping Chen & Yi Chai & Ranchao Wu, 2011. "Control and Synchronization of Fractional-Order Financial System Based on Linear Control," Discrete Dynamics in Nature and Society, Hindawi, vol. 2011, pages 1-21, August.
    26. Hagen Kleinert & Jan Korbel, 2015. "Option Pricing Beyond Black-Scholes Based on Double-Fractional Diffusion," Papers 1503.05655, arXiv.org, revised Mar 2016.
    27. Álvaro Cartea & Thilo Meyer-Brandis, 2010. "How Duration Between Trades of Underlying Securities Affects Option Prices," Review of Finance, European Finance Association, vol. 14(4), pages 749-785.
    28. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    29. Kleinert, H. & Korbel, J., 2016. "Option pricing beyond Black–Scholes based on double-fractional diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 449(C), pages 200-214.
    30. Álvaro Cartea, 2013. "Derivatives pricing with marked point processes using tick-by-tick data," Quantitative Finance, Taylor & Francis Journals, vol. 13(1), pages 111-123, January.
    31. Meerschaert, Mark M. & Scalas, Enrico, 2006. "Coupled continuous time random walks in finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(1), pages 114-118.
    32. Škovránek, Tomáš & Podlubny, Igor & Petráš, Ivo, 2012. "Modeling of the national economies in state-space: A fractional calculus approach," Economic Modelling, Elsevier, vol. 29(4), pages 1322-1327.
    33. Cartea, Álvaro & del-Castillo-Negrete, Diego, 2007. "Fractional diffusion models of option prices in markets with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 374(2), pages 749-763.
    34. Leeson,Robert (ed.), 2000. "A. W. H. Phillips: Collected Works in Contemporary Perspective," Cambridge Books, Cambridge University Press, number 9780521571357, September.
    35. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
    36. Hajipour, Ahamad & Hajipour, Mojtaba & Baleanu, Dumitru, 2018. "On the adaptive sliding mode controller for a hyperchaotic fractional-order financial system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 497(C), pages 139-153.
    37. Valentina V. Tarasova & Vasily E. Tarasov, 2017. "Economic Growth Model with Constant Pace and Dynamic Memory," Papers 1701.06299, arXiv.org, revised Apr 2019.
    38. David, S.A. & Machado, J.A.T. & Quintino, D.D. & Balthazar, J.M., 2016. "Partial chaos suppression in a fractional order macroeconomic model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 122(C), pages 55-68.
    39. Baillie, Richard T., 1996. "Long memory processes and fractional integration in econometrics," Journal of Econometrics, Elsevier, vol. 73(1), pages 5-59, July.
    40. Yiding Yue & Lei He & Guanchun Liu, 2013. "Modeling and Application of a New Nonlinear Fractional Financial Model," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-9, November.
    41. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    42. Arkadii A. Chikrii, 2008. "Game Dynamic Problems for Systems with Fractional Derivatives," Springer Optimization and Its Applications, in: Altannar Chinchuluun & Panos M. Pardalos & Athanasios Migdalas & Leonidas Pitsoulis (ed.), Pareto Optimality, Game Theory And Equilibria, pages 349-386, Springer.
    43. Stefan Rostek, 2009. "Option Pricing in Fractional Brownian Markets," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-642-00331-8, October.
    44. Stefan Rostek, 2009. "Risk Preference Based Option Pricing in a Continuous Time Fractional Brownian Market," Lecture Notes in Economics and Mathematical Systems, in: Option Pricing in Fractional Brownian Markets, chapter 5, pages 79-110, Springer.
    45. Mohammed Salah Abd-Elouahab & Nasr-Eddine Hamri & Junwei Wang, 2010. "Chaos Control of a Fractional-Order Financial System," Mathematical Problems in Engineering, Hindawi, vol. 2010, pages 1-18, July.
    46. Rostek, S. & Schöbel, R., 2013. "A note on the use of fractional Brownian motion for financial modeling," Economic Modelling, Elsevier, vol. 30(C), pages 30-35.
    47. Gilles Teyssière & Alan P. Kirman (ed.), 2007. "Long Memory in Economics," Springer Books, Springer, number 978-3-540-34625-8, December.
    48. Vasily E. Tarasov & Valentina V. Tarasova, 2017. "Accelerators in Macroeconomics: Comparison of Discrete and Continuous Approaches," American Journal of Economics and Business Administration, Science Publications, vol. 9(3), pages 47-55, November.
    49. Valentina V. Tarasova & Vasily E. Tarasov, 2017. "Logistic map with memory from economic model," Papers 1712.09092, arXiv.org.
    50. Luis A. Gil-Alana & Javier Hualde, 2009. "Fractional Integration and Cointegration: An Overview and an Empirical Application," Palgrave Macmillan Books, in: Terence C. Mills & Kerry Patterson (ed.), Palgrave Handbook of Econometrics, chapter 10, pages 434-469, Palgrave Macmillan.
    51. C. W. J. Granger & Roselyne Joyeux, 1980. "An Introduction To Long‐Memory Time Series Models And Fractional Differencing," Journal of Time Series Analysis, Wiley Blackwell, vol. 1(1), pages 15-29, January.
    52. Dadras, Sara & Momeni, Hamid Reza, 2010. "Control of a fractional-order economical system via sliding mode," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(12), pages 2434-2442.
    53. William R. Parke, 1999. "What Is Fractional Integration?," The Review of Economics and Statistics, MIT Press, vol. 81(4), pages 632-638, November.
    54. Solow, Robert M., 1999. "Neoclassical growth theory," Handbook of Macroeconomics, in: J. B. Taylor & M. Woodford (ed.), Handbook of Macroeconomics, edition 1, volume 1, chapter 9, pages 637-667, Elsevier.
    55. Nicholas Kaldor, 1961. "Capital Accumulation and Economic Growth," International Economic Association Series, in: D. C. Hague (ed.), The Theory of Capital, chapter 0, pages 177-222, Palgrave Macmillan.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Vasily E. Tarasov, 2024. "General Fractional Economic Dynamics with Memory," Mathematics, MDPI, vol. 12(15), pages 1-24, August.
    2. Muhammad Imran Liaqat & Ali Akgül & Hanaa Abu-Zinadah, 2023. "Analytical Investigation of Some Time-Fractional Black–Scholes Models by the Aboodh Residual Power Series Method," Mathematics, MDPI, vol. 11(2), pages 1-19, January.
    3. Kyriaki Tsilika, 2023. "Exploring the Contributions to Mathematical Economics: A Bibliometric Analysis Using Bibliometrix and VOSviewer," Mathematics, MDPI, vol. 11(22), pages 1-21, November.
    4. Arsen Pskhu, 2020. "Green Functions of the First Boundary-Value Problem for a Fractional Diffusion—Wave Equation in Multidimensional Domains," Mathematics, MDPI, vol. 8(4), pages 1-15, March.
    5. İbrahim Avcı & Nazim I. Mahmudov, 2020. "Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration," Mathematics, MDPI, vol. 8(1), pages 1-24, January.
    6. Vasily E. Tarasov, 2020. "Non-Linear Macroeconomic Models of Growth with Memory," Mathematics, MDPI, vol. 8(11), pages 1-22, November.
    7. Arsen Pskhu & Sergo Rekhviashvili, 2020. "Fractional Diffusion–Wave Equation with Application in Electrodynamics," Mathematics, MDPI, vol. 8(11), pages 1-13, November.
    8. Turkyilmazoglu, Mustafa & Altanji, Mohamed, 2023. "Fractional models of falling object with linear and quadratic frictional forces considering Caputo derivative," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    9. Ertuğrul Karaçuha & Vasil Tabatadze & Kamil Karaçuha & Nisa Özge Önal & Esra Ergün, 2020. "Deep Assessment Methodology Using Fractional Calculus on Mathematical Modeling and Prediction of Gross Domestic Product per Capita of Countries," Mathematics, MDPI, vol. 8(4), pages 1-18, April.
    10. Ohud Almutairi & Adem Kılıçman, 2020. "Integral Inequalities for s -Convexity via Generalized Fractional Integrals on Fractal Sets," Mathematics, MDPI, vol. 8(1), pages 1-11, January.
    11. Wang, Bo & Liu, Jinping & Alassafi, Madini O. & Alsaadi, Fawaz E. & Jahanshahi, Hadi & Bekiros, Stelios, 2022. "Intelligent parameter identification and prediction of variable time fractional derivative and application in a symmetric chaotic financial system," Chaos, Solitons & Fractals, Elsevier, vol. 154(C).
    12. Shi, Jianping & He, Ke & Fang, Hui, 2022. "Chaos, Hopf bifurcation and control of a fractional-order delay financial system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 348-364.
    13. Vasily E. Tarasov, 2020. "Mathematical Economics: Application of Fractional Calculus," Mathematics, MDPI, vol. 8(5), pages 1-3, April.
    14. Monica Aureliana Petcu & Liliana Ionescu-Feleaga & Bogdan-Ștefan Ionescu & Dumitru-Florin Moise, 2023. "A Decade for the Mathematics : Bibliometric Analysis of Mathematical Modeling in Economics, Ecology, and Environment," Mathematics, MDPI, vol. 11(2), pages 1-30, January.
    15. Borin, Daniel, 2024. "Caputo fractional standard map: Scaling invariance analyses," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    16. José A. Tenreiro Machado & Maria Eugénia Mata & António M. Lopes, 2020. "Fractional Dynamics and Pseudo-Phase Space of Country Economic Processes," Mathematics, MDPI, vol. 8(1), pages 1-17, January.
    17. Hou, Jie & Ma, Zhiying & Ying, Shihui & Li, Ying, 2024. "HNS: An efficient hermite neural solver for solving time-fractional partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    18. He, Ke & Shi, Jianping & Fang, Hui, 2024. "Bifurcation and chaos analysis of a fractional-order delay financial risk system using dynamic system approach and persistent homology," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 223(C), pages 253-274.
    19. Jean-Philippe Aguilar & Jan Korbel & Yuri Luchko, 2019. "Applications of the Fractional Diffusion Equation to Option Pricing and Risk Calculations," Mathematics, MDPI, vol. 7(9), pages 1-23, September.
    20. Inés Tejado & Emiliano Pérez & Duarte Valério, 2020. "Fractional Derivatives for Economic Growth Modelling of the Group of Twenty: Application to Prediction," Mathematics, MDPI, vol. 8(1), pages 1-21, January.
    21. Bazán Navarro, Ciro Eduardo & Benazic Tomé, Renato Mario, 2024. "Qualitative behavior in a fractional order IS-LM-AS macroeconomic model with stability analysis," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 217(C), pages 425-443.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Valentina V. Tarasova & Vasily E. Tarasov, 2017. "Concept of dynamic memory in economics," Papers 1712.09088, arXiv.org.
    2. Vasily E. Tarasov, 2019. "Rules for Fractional-Dynamic Generalizations: Difficulties of Constructing Fractional Dynamic Models," Mathematics, MDPI, vol. 7(6), pages 1-50, June.
    3. Tarasov, Vasily E. & Tarasova, Valentina V., 2018. "Macroeconomic models with long dynamic memory: Fractional calculus approach," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 466-486.
    4. Vasily E. Tarasov & Valentina V. Tarasova, 2019. "Dynamic Keynesian Model of Economic Growth with Memory and Lag," Mathematics, MDPI, vol. 7(2), pages 1-17, February.
    5. Tarasov, Vasily E., 2020. "Fractional econophysics: Market price dynamics with memory effects," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 557(C).
    6. Shi, Jianping & He, Ke & Fang, Hui, 2022. "Chaos, Hopf bifurcation and control of a fractional-order delay financial system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 348-364.
    7. Vasily E. Tarasov, 2020. "Non-Linear Macroeconomic Models of Growth with Memory," Mathematics, MDPI, vol. 8(11), pages 1-22, November.
    8. Ali Balcı, Mehmet, 2017. "Time fractional capital-induced labor migration model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 477(C), pages 91-98.
    9. Valentina V. Tarasova & Vasily E. Tarasov, 2017. "Dynamic intersectoral models with power-law memory," Papers 1712.09087, arXiv.org.
    10. Jajarmi, Amin & Hajipour, Mojtaba & Baleanu, Dumitru, 2017. "New aspects of the adaptive synchronization and hyperchaos suppression of a financial model," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 285-296.
    11. Valentina V. Tarasova & Vasily E. Tarasov, 2016. "Fractional Dynamics of Natural Growth and Memory Effect in Economics," Papers 1612.09060, arXiv.org, revised Jan 2017.
    12. Valentina V. Tarasova & Vasily E. Tarasov, 2017. "Logistic map with memory from economic model," Papers 1712.09092, arXiv.org.
    13. Tarasova, Valentina V. & Tarasov, Vasily E., 2017. "Logistic map with memory from economic model," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 84-91.
    14. Hajipour, Ahamad & Hajipour, Mojtaba & Baleanu, Dumitru, 2018. "On the adaptive sliding mode controller for a hyperchaotic fractional-order financial system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 497(C), pages 139-153.
    15. Jean-Philippe Aguilar & Jan Korbel & Yuri Luchko, 2019. "Applications of the Fractional Diffusion Equation to Option Pricing and Risk Calculations," Mathematics, MDPI, vol. 7(9), pages 1-23, September.
    16. G. Fern'andez-Anaya & L. A. Quezada-T'ellez & B. Nu~nez-Zavala & D. Brun-Battistini, 2019. "Katugampola Generalized Conformal Derivative Approach to Inada Conditions and Solow-Swan Economic Growth Model," Papers 1907.00130, arXiv.org.
    17. Scalas, Enrico & Politi, Mauro, 2012. "A parsimonious model for intraday European option pricing," Economics Discussion Papers 2012-14, Kiel Institute for the World Economy (IfW Kiel).
    18. David, S.A. & Machado, J.A.T. & Quintino, D.D. & Balthazar, J.M., 2016. "Partial chaos suppression in a fractional order macroeconomic model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 122(C), pages 55-68.
    19. Scalas, Enrico, 2007. "Mixtures of compound Poisson processes as models of tick-by-tick financial data," Chaos, Solitons & Fractals, Elsevier, vol. 34(1), pages 33-40.
    20. Torricelli, Lorenzo, 2020. "Trade duration risk in subdiffusive financial models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 541(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:7:y:2019:i:6:p:509-:d:237116. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.