Volume 11, no. 2Pages 58 - 72

Stochastic Leontief Type Equations with Impulse Actions

E.Yu. Mashkov
By a stochastic Leontief type equation we mean a special class of stochastic differential equations in the Ito form, in which there is a degenerate constant linear operator in the left-hand side and a non-degenerate constant linear operator in the right-hand side. In addition, in the right-hand side there is a deterministic term that depends only on time, as well as impulse effects. It is assumed that the diffusion coefficient of this system is given by a square matrix, which depends only on time. To study the equations under consideration, it is required to consider derivatives of sufficiently high orders from the free terms, including the Wiener process. In connection with this, to differentiate the Wiener process, we apply the machinery of Nelson mean derivatives of random processes, which makes it possible to avoid using the theory of generalized functions to the study of equations. As a result, analytical formulas are obtained for solving the equation in terms of mean derivatives of random processes.
Full text
mean derivative; current velocity; Wiener process; stochastic Leontief type equation.
1. Shestakov A.L., Sviridyuk G.A. A New Approach to the Measurement of Dynamically Distorted Signals. Bulletin of South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2010, no. 16 (192), issue 5, pp. 116-120.
2. Shestakov A.L., Keller A.V., Sviridyuk G.A. The Theory of Optimal Measurements. Journal of Computational and Engineering Mathematics, 2014, vol. 1, no. 1, pp. 3-16.
3. Shestakov A.L., Sviridyuk G.A. On the Measurement of the 'White Noise'. Bulletin of South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 27 (286), issue 13, pp. 99-108.
4. Schein O., Denk G. Numerical Solution of Stochastic Differential-Algebraic Equations with Applications to Transient Noise Simulation of Microelectronic Circuits. Journal of Computational and Applied Mathematics, 1998, vol. 100, no. 1, pp. 77-92. DOI: 10.1016/S0377-0427(98)00138-1
5. Sickenberger T., Winkler R. Stochastic Oscillations in Circuit Simulation. Proceeding in Applied Mathematics and Mechanics, 2007, vol. 7, no. 1, pp. 4050023-4050024. DOI: 10.1002/pamm.200700807
6. Winkler R. Stochastic DAEs in Transient Noise Simulation. Proceedings of Scientific Computing in Electrical Engineering, 2004, vol. 4, pp. 408-415. DOI: 10.1007/978-3-642-55872-6_45
7. Vlasenko L.A., Lysenko Yu.G., Rutkas A.G. About One Stochastic Model of Enterprise Corporations Dynamics. Economic Cybernetics, 2011, no. 1-3 (67-69), pp. 4-9.
8. Vlasenko L.A., Lyshko S.L., Rutkas A.G. On a Stochastic Impulsive System. Reports of the National Academy of Sciences of Ukraine, 2012, no. 2, pp. 50-55.
9. Belov A.A., Kurdyukov A.P. [Descriptor Systems and Control Problems]. Moscow, Fizmatlit, 2015. (in Russian)
10. Mashkov E.Yu. Stochastic Equations of Leontief Type with Time-Dependent Diffusion Coefficient. Bulletin of Voronezh State University. Series: Physics. Mathematics, 2017, no. 3, pp. 148-158.
11. Mashkov E.Yu. Singular Stochastic Leontieff Type Equation with Dependency on Time Diffusion Coefficients. Global and Stochastic Analysis, 2017, vol. 4, no. 2, pp. 207-217.
12. Nelson E. Derivation of the Schr'odinger Equation from Newtonian Mechanics. Physics Reviews, 1966, vol. 150, no. 4, pp. 1079-1085. DOI: 10.1103/PhysRev.150.1079
13. Nelson E. Dynamical Theory of Brownian Motion. Princeton, Princeton University Press, 1967.
14. Nelson E. Quantum Fluctuations. Princeton, Princeton University Press, 1985.
15. Gliklikh Yu.E. Global and Stochastic Analysis in Mathematical Problems Physics. Moscow, Comkniga, 2005.
16. Gliklikh Yu.E., Mashkov E.Yu. Stochastic Leontieff Type Equation with Non-Constant Coefficients. Applicable Analysis: An International Journal, 2015, vol. 94, no. 8, pp. 1614-1623. DOI: 10.1080/00036811.2014.940521
17. Gliklikh Yu.E., Mashkov E.Yu. Stochastic Leontief Type Equations and Derivatives in the Mean of Stochastic Processes. Bulletin of South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2013, vol. 6, no. 2, pp.25-39.
18. Partasarati K.R. Introduction to Probability Theory and Measure Theory. Moscow, Mir, 1988.
19. Gantmakher F.R. The Matrix Theory. Moscow, Fizmatlit, 1967.
20. Gihman I.I., Scorohod A.V. Theory of Stochastic Processes, N.Y., Springer, 1979.