Stochastic calculation of curves dynamics of enterprise
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 24 (2020) no. 2, pp. 343-364.

Voir la notice de l'article provenant de la source Math-Net.Ru

The article proposes mathematical models of the stochastic dynamics of the single-factor manufacturing enterprises development through internal and external investments. Balance equations for such enterprises are formulated, describing random processes of continuous increase in output and growth of production factors. The interaction of proportional, progressive and digressive depreciation with internal and external investments is investigated. Equations are obtained to determine the equilibrium state of the enterprise and the limiting values of the factors of production are calculated. The cases of the stable progressive development of the enterprise, the suspension of its work during the re-equipment of production and the temporary crisis of production shutdown during equipment replacement are considered. The algorithm for the numerical solution of stochastic differential equations of enterprise development is constructed in accordance with the Euler–Maruyama method. For each implementation of this algorithm, the corresponding stochastic trajectories are constructed for the random function of the production factor. A variant of the method for calculating the expectation of a random function of a factor of production is developed and the corresponding differential equation is obtained for it. It is shown that the numerical solution of this equation and the average value of the function of the production factor calculated from two hundred realizations of stochastic trajectories give almost identical results. Numerical analysis of the developed models showed good compliance with the known statistical data of the production enterprise.
Keywords: production factors, production function, stochastic equations, Wiener process, drift coefficient, volatility factor, Euler–Maruyama method.
@article{VSGTU_2020_24_2_a6,
     author = {A. L. Saraev and L. A. Saraev},
     title = {Stochastic calculation of curves dynamics of enterprise},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {343--364},
     publisher = {mathdoc},
     volume = {24},
     number = {2},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2020_24_2_a6/}
}
TY  - JOUR
AU  - A. L. Saraev
AU  - L. A. Saraev
TI  - Stochastic calculation of curves dynamics of enterprise
JO  - Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
PY  - 2020
SP  - 343
EP  - 364
VL  - 24
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VSGTU_2020_24_2_a6/
LA  - ru
ID  - VSGTU_2020_24_2_a6
ER  - 
%0 Journal Article
%A A. L. Saraev
%A L. A. Saraev
%T Stochastic calculation of curves dynamics of enterprise
%J Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
%D 2020
%P 343-364
%V 24
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VSGTU_2020_24_2_a6/
%G ru
%F VSGTU_2020_24_2_a6
A. L. Saraev; L. A. Saraev. Stochastic calculation of curves dynamics of enterprise. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 24 (2020) no. 2, pp. 343-364. http://geodesic.mathdoc.fr/item/VSGTU_2020_24_2_a6/

[1] Kolemayev V. A., Matematicheskaia ekonomika [Mathematical Economics], Moscow, 2005, 399 pp. (In Russian)

[2] Artemyev S. S., Yakunin M. A., Matematicheskoe i statisticheskoe modelirovanie v finansakh [Mathematical and Statistical Modeling in Finances], Novosibirsk, 2008, 174 pp. (In Russian)

[3] Vorontsovskii A. V., Dikarev A. Yu., “Forecasting macroeconomic indicators in simulation mode based on stochastic models of economic growth”, Finansy i Biznes, 2013, no. 2, 33–51 (In Russian)

[4] Kurzenev V. A., Lychagina E. B., “Stochastic Modelling of Dynamics of Economic System”, Upravlencheskoe konsultirovanie, 2013, no. 5, 78–83 (In Russian)

[5] Andrianov D. L., Shultz D. N., Oshchepkov I. A., “Dynamic stochastic general economic equilibrium models”, Upravlenie ekonomicheskimi sistemami [Management of Economic Systems], 67:7 (2014) (In Russian) http://www.uecs.ru/uecs67-672014/item/2998-2014-07-30-07-14-51

[6] Andrianov D. L., Shultz D. N., Oshchepkov I. A., “Dynamic stochastic model of Russia's general economic equilibrium”, Vestnik Nizhegorodskogo universiteta, Ser. Sotsialnye nauki, 2015, no. 2(38), 18–25 (In Russian)

[7] Andrianov D. L., Arbuzov V. O., Ivliev S. V., Maksimov V. P., Simonov P. M., “Dynamic models of economics: Theory, applications, software implementation”, Vestnik Permskogo Universiteta, Ser. Ekonomika, 2015, no. 4, 8–32

[8] Itô K., McKean H. P. Jr., Diffusion processes and their sample paths, Classics in Mathematics, Springer, Berlin, xv+321 pp. | DOI

[9] Allen E., Modeling with Itô stochastic differential equations, Mathematical Modelling: Theory and Applications, 22, Springer, Netherlands, 2007, xii+230 pp. | DOI | Zbl

[10] Stepanov S. S., Stokhasticheskii mir [Stochastic World] ; Accessed February 27, 2020 (In Russian) https://synset.com/pdf/ito.pdf

[11] Neisy A., Peymany M., “Financial modeling by ordinary and stochastic differential equations”, World Applied Sciences Journal, 13:11 (2011), 2288–2295

[12] Kallianpur G., Sundar P., Stochastic analysis and diffusion processes, Oxford Graduate Texts in Mathematics, 24, Oxford University Press, Oxford, 2014, xiv+352 pp. | DOI | Zbl

[13] Bally V., Talay D., “The law of the Euler scheme for stochastic differential equations: I. Convergence rate of the distribution function”, Probab. Th. Rel. Fields, 104:1 (1996), 43–60 | DOI | Zbl

[14] Bally V., Talay D., “The law of the Euler scheme for stochastic differential equations: II. Convergence rate of the density”, Monte Carlo Methods and Applications, 2:2 (1996), 93–128 | DOI | Zbl

[15] Debarant K., Rößler A., “Classification of stochastic Runge–Kutta methods for the weak approximation of stochastic differential equations”, Mathematics and Computers in Simulation, 77:4 (2008), 408–420, arXiv: [math.NA] 1303.4510 | DOI

[16] Soheili A. R., Namjoo M., “Strong approximation of stochastic differential equations with Runge–Kutta methods”, World Journal of Modelling and Simulation, 4:2 (2008), 83–93

[17] Kuznetsov D. S., Stokhasticheskie differentsial'nye uravneniia: teoriia i praktika chislennogo resheniia [Stochastic Differential Equations: Theory and Practice of Numerical Solution], Polytechnic Univ., St. Petersburg, 2007, 800 pp. (In Russian) | DOI

[18] Konakov V., Menozzi S., “Weak error for stable driven stochastic differential equations: Expansion of the densities”, J. Theor. Probab., 24 (2011), 454–478 | DOI | Zbl

[19] Konakov V., Menozzi S., “Weak error for the Euler scheme approximation of diffusions with non-smooth coefficients”, Electron. J. Probab., 22 (2017), 46, 47 pp., arXiv: [math.PR] 1604.00771 | DOI | Zbl

[20] Hottovy S., Volpe G., Wehr J., “Noise-Induced drift in stochastic differential equations with arbitrary friction and diffusion in the Smoluchowski–Kramers limit”, J. Stat. Phys., 146:4 (2012), 762–773 | DOI | Zbl

[21] Frikha N., “On the weak approximation of a skew diffusion by an Euler-type scheme”, Bernoulli, 24:3 (2018), 1653–1691 | DOI | Zbl

[22] Solov'ev V. I., Ekonomiko-matematicheskoe modelirovanie rynka programmnogo obespecheniia [Economic and Mathematical Modeling of the Software Market], Vega-Info, Moscow, 2009, 176 pp. (In Russian)

[23] Ilyina E. A., Saraev A. L., Saraev L. A., “To the theory of modernization of manufacturing enterprises, taking into account the lag of domestic investment”, Ekonomika i predprinimatel'stvo, 2017, no. 9–4(86), 1130–1134 (In Russian)

[24] Kuznetzova I. Yu., “Numerical solution of a stochastic differential equation by the Euler–Maruyama method”, Mezhdunarodnyi nauchno-issledovatel'skii zhurnal, 2013, no. 11–1(18), 8–11 (In Russian)

[25] Accounting (financial) statements for PJSC Chelyabinsk Tube Rolling Plant after 2017, ; Accessed February 27, 2020 (In Russian) https://e-ecolog.ru/buh/2017/7449006730

[26] Accounting (financial) statements for LADA Izhevsk Automobile Plant LLC after 2017, ; Accessed February 27, 2020 (In Russian) https://e-ecolog.ru/buh/2017/1834051678

[27] Saraev A. L., Saraev L. A., “Indicators of nonlinear dynamics and the limiting condition of a manufacturing enterprise”, Ekonomika i predprinimatel'stvo, 2018, no. 11, 1237–1241 (In Russian)