Macroeconomic models with long dynamic memory: Fractional calculus approach

Tarasov, Vasily E.; Tarasova, Valentina V.

doi:10.1016/j.amc.2018.06.018

Cited by 58 publications

(67 citation statements)

References 44 publications

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“…During the last decades FC has become a popular tool [29][30][31], and new areas of application have emerged. The modeling finance [32] and economy [18,21,22,26,[33][34][35] phenomena became of particular relevance.…”

Section: The Fractional Calculusmentioning

confidence: 99%

Fractional Dynamics and Pseudo-Phase Space of Country Economic Processes

2020

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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.

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Section: The Fractional Calculusmentioning

confidence: 99%

Fractional Dynamics and Pseudo-Phase Space of Country Economic Processes

2020

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“…where n − 1 < α ≤ n. Solution (16) of the fractional differential Equation (14) and its properties are described in [30,31] (see also [32][33][34]).…”

Section: Dynamic Keynesian Model With Memorymentioning

confidence: 99%

“…An application of advanced mathematical methods of fractional calculus in Keynesian economic models with continuous time was proposed by authors [30,31] in 2016 (see also [32][33][34]). The fractional differential equations of the dynamic Keynesian model with power-law memory and their solutions have been considered in [30][31][32][33][34]. Continuously distributed lag was not discussed in these works.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Keynesian Model of Economic Growth with Memory and Lag

Tarasova

2019

Mathematics

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A mathematical model of economic growth with fading memory and continuous distribution of delay time is suggested. This model can be considered as a generalization of the standard Keynesian macroeconomic model. To take into account the memory and gamma-distributed lag we use the Abel-type integral and integro-differential operators with the confluent hypergeometric Kummer function in the kernel. These operators allow us to propose an economic accelerator, in which the memory and lag are taken into account. The fractional differential equation, which describes the dynamics of national income in this generalized model, is suggested. The solution of this fractional differential equation is obtained in the form of series of the confluent hypergeometric Kummer functions. The asymptotic behavior of national income, which is described by this solution, is considered.

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“…Now, it follows from the reflection formula for the Gamma function that one can re-write the FMLS risk-neutral parameter (22) as follows:…”

Section: Risk-neutral Parametermentioning

confidence: 99%

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

2019

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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. IntroductionFractional Calculus (FC) is nearly as old as conventional calculus. Many prominent mathematicians including Leibniz, Fourier, Laplace, Liouville, Riemann, Weyl, and Riesz suggested their own definitions of the fractional integrals and derivatives and studied their properties. Whereas the mathematical theory of FC was nearly completed a long time ago (see, e.g., [1] or the recent reference books [2,3]), only a few applications of FC outside mathematics were known until recently. During the last three decades, the situation changed completely, and currently, the majority of FC publications is devoted to modeling of a broad class of systems and processes using either the FC operators or the so-called fractional ordinary or partial differential equations. In particular, the FC models were successfully employed in physics [4,5], control theory [6], as well as in engineering, life, and social sciences [7,8], to mention only a few of the many application areas.The two probably most prominent and broadly-recognized FC applications are in linear viscoelasticity [9] and for describing anomalous transport processes [10]. In both cases, the FC models in the form of the fractional ODEs (linear viscoelasticity) and the fractional PDEs (anomalous transport processes), respectively, cannot be derived from "first principles". Instead, they are introduced as interpolations between several models formulated in terms of the ODEs or PDEs, respectively. In the case of linear viscoelasticity, the basic FC model interpolates between the Hooke model (elasticity, ODE Mathematics 2019, 7, 796 2 of 23 of the zero order) and the Newton model (viscosity, ODE of the first order). The FC model for the slow anomalous diffusion interpolates between a stationary state (time-independent Poisson equation) and the diffusion equation (first order equation with respect to time), whereas the fast diffusion is modeled by the fractional diffusion-wave PDE that interpolates between the diffusion equation (first order equation with respect to time) and the wave equation (second order equation with respect to time).Of course, this kind of model needs an additional justification...

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Macroeconomic models with long dynamic memory: Fractional calculus approach

Cited by 58 publications

References 44 publications

Fractional Dynamics and Pseudo-Phase Space of Country Economic Processes

Fractional Dynamics and Pseudo-Phase Space of Country Economic Processes

Dynamic Keynesian Model of Economic Growth with Memory and Lag

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

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