1. MMC
  2. All Volumes and Issues
  3. Volume 6, Number 1, 2019
  4. Generalized Fokker–Planck equation for the distribution function of liquidity accumulation

Generalized Fokker–Planck equation for the distribution function of liquidity accumulation

MMC.
2019;
Volume 6, Number 1
: pp. 37-43
https://doi.org/10.23939/mmc2019.01.037
Received: March 03, 2019
Revised: April 23, 2019
Accepted: April 30, 2019

Math. Model. Comput. Vol.6, No.1, pp.37-43 (2019)

Authors:
  1. B. Hnativ
  2. A. Didyk
  3. M. Tokarchuk
1
Lviv Polytechnic National University
2
Lviv Polytechnic National University
3
Lviv Polytechnic National University; Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine

By means of the method of Zubarev's nonequilibrium statistical operator, the generalized Fokker--Planck equation for the distribution function of liquidity accumulations has been obtained.  The generalized velocity and transport kernels describing dynamic correlations between liquidity accumulations of different categories of families have been determined.  The system of non-Markov transport equations for non-equilibrium average values of liquidity accumulations for different categories of families has been obtained.  Memory effects have been analyzed using the fractional calculus, which has led to a system of transport equations for non-equilibrium average values of liquidity accumulations for different categories of families in fractional derivatives.

transport equation
memory effects
liquidity accumulations
fractional derivative
  1. LaRouche L.  The Science of Physical Economy as the Platonic Epistemological Basis for All Branches of Human Knowledge.  Executive Intelligence Review. 21 (9--11), (1994).
  2. Mantegna R. N., Stanley H. E.  An introduction to econophysics. Correlations and complexity in finance.  Cambridge, Cambridge University Press (1999).
  3. Bouchaud J.-P., Potters M.  Theory of financial risks.  Cambridge, Cambridge University Press (2000).
  4. Voit J.  The statistical mechanics of financial markets.  Berlin--Heidelberg, Springer--Verlag (2001).
  5. Mayevsky V. I.  Introduction to evolutionary economics.  Moscow, Japan today (2000), (in Russian).
  6. Nelson R. R., Winter S. J.  Evolutionary theory of economic change.  Moscow, Delo (2000), (in Russian).
  7. Pu T.  Nonlinear economic dynamic.  Izhevsk, Udmurt. university (2000), (in Russian).
  8. Zhang V.-B.  Synergetic economics: Time and changes in non-linear economic theory.  Berlin--Heidelberg, Springer (1991).
  9. Lebedev V. V.  Mathematical modeling of socio-economic processes.  Moscow, Isograph (1997), (in Russian).
  10. Olemskoi A. I., Yushchenko O. V.  The synergetic picture of the financial market, evolving in accordance with the incoming information.  Economic regulation mechanism. 1, 112--117 (2003), (in Russian).
  11. Chernavskii D. S., Starkov N. I., Shcherbakov A. V.  On some problems of physical economics.  Physics-Uspekhi. 45 (9), 977--997 (2002).
  12. Yukhnovsky I. R.  The program of the future. Selected Works. Economics.  Lviv Polytechnic National University, pp.\,3--18 (2005), (in Ukrainian).
  13. Hnativ B. V., Tokarchuk R. M., Kostrobij P. P., Tokarchuk M. V.  Mathematical modeling of economic processes by method of nonequilibrium statistical mechanics.  J. Lviv Polytechnic National University, Physical mathematical sciences. 696, 93--100 (2011), (in Ukrainian).
  14. Dmytryshyn L. I.  Ecophysical aspects of the formation of the function of distribution of wealth and income in socio-economic systems.  Modeling of Regional Economics: Proceedings. 1, 96-–102 (2012), (in Ukrainian).
  15. Dmytryshyn L. I.  Kinetic models of distribution of money income of the population.  Modeling of Regional Economics: Proceedings. 2, 50--62 (2012), (in Ukrainian).
  16. Moroz K. V.  Income distribution of the population of Ukraine: Empirical analysis using lognormal function.  Vіsnik Harkіvs'kogo nacіonal'nogo unіversitetu іmenі V. N. Karazіna. Serіja "Ekonomіchna''. 91, 110--117 (2016), (in Ukrainian).
  17. Baillie R. T.  Long memory processes and fractional integration in econometrics.  J. Econometrics. 73, 5--59 (1996).
  18. Scalas E., Gorenflo R., Mainardi F.  Fractional calculus and continuous-time finance.  Physica A. 284, 378--384 (2000).
  19. Mainardi F., Raberto M., Gorenflo R., Scalas E.  Fractional calculus and continuous-time finance II: The waiting-time distribution.  Physica A. 287, 468--481 (2000).
  20. Laskin N.  Fractional market dynamics.  Physica A. 287, 482--492 (2000).
  21. Gorenflo R., Mainardi F., Scalas E., Raberto M.  Fractional calculus and continuous-time finance III: the diffusion limit.  Mathematical Finance. 171--180 (2001).
  22. Raberto M., Scalas E., Mainardi F.  Waiting-times and returns in high-frequency financial data: an empirical study.  Physica A. 314, 749--755 (2002).
  23. Teyssiere G., Kirman A. P.  Long Memory in Economics.  Berlin--Heidelberg, Springer--Verlag (2007).
  24. Škovránek T., Podlubny I., Petráš I.  Modeling of the national economies in state-space: A fractional calculus approach.  Economic Modelling. 29 (4), 1322--1327 (2012).
  25. Tarasova V. V., Tarasov V. E.  Elasticity for economic processes with memory: fractional differential calculus approach.  Fractional Differential Calculus. 6 (2), 219--232 (2016).
  26. Tarasov V. E., Tarasova V. V.  Time-dependent fractional dynamics with memory in quantum and economic physics.  Ann.  Phys. 383, 579--599 (2017).
  27. Tarasova V. V., Tarasov V. E.  Economic interpretation of fractional derivatives.  Prog. Fractional Diff. Appl. 3 (1), 1--7 (2017).
  28. Tarasov V. E., Tarasova V. V.  Macroeconomic models with long dynamic memory: Fractional calculus approach.  Appl. Math. Comp. 338, 466--486 (2018).
  29. Tarasova V. V., Tarasov V. E.  Concept of dynamics memory in economics.  Comm. Nonlin. Sci. Numer. Simul. 55, 127--145 (2018).
  30. Samko S. G., Kilbas A. A., Marichev O. I.  Fractional Integrals and Derivatives: Theory and Applications.  New York, Gordon and Breach (1993).
  31. Podlubny I.  Fractional Differential Equations.  San Diego, Academic Press (1998).
  32. Kilbas A. A., Srivastava H. M., Trujillo J. J.  Theory and Applications of Fractional Differential Equations.  Amsterdam, Elsevier (2006).
  33. Diethelm K.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type.  Berlin, Springer--Verlag (2010).
  34. Zubarev D. N., Khazanov A. M.  Generalized Fokker–-Planck equation and construction of projection operators for different methods of reduced description of nonequilibrium states. Theor. Math. Phys. 34 (1), 43--50 (1978).
  35. Zubarev D. N.,  Morozov V. G., Röpke G.  Statistical mechanics of nonequilibrium processes, vol. 2.  Moscow, Fizmatlit (2002), (in Russian).
  36. Hlushak P. A., Tokarchuk M. V.  Chain of kinetic equations for the distribution functions of particles in simple liquid taking into account nonlinear hydrodynamic fluctuations.  Physica A. 443, 231--245 (2016).
  37. Kostrobij P. P., Markovych B. M., Viznovych O. V., Tokarchuk M. V.  Generalized transport equation with nonlocality of space–time. Zubarev’s NSO method.  Physica A. 514, 63--70 (2019).