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Stochastic Leontief type equations with impulse actions. (English) Zbl 1402.60084

Summary: 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.

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[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, 16 (192):5 (2010), 116–120 · Zbl 1243.94015
[2] A.L. Shestakov, A.V. Keller, G.A. Sviridyuk, “The Theory of Optimal Measurements”, Journal of Computational and Engineering Mathematics, 1:1 (2014), 3–16 · Zbl 1343.49005
[3] A.L. Shestakov, G.A. Sviridyuk, “On the Measurement of the «White Noise»”, Bulletin of South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 27 (286):13 (2012), 99–108 · Zbl 1413.60065
[4] O. Schein, G. Denk, “Numerical Solution of Stochastic Differential-Algebraic Equations with Applications to Transient Noise Simulation of Microelectronic Circuits”, Journal of Computational and Applied Mathematics, 100:1 (1998), 77–92 · Zbl 0928.65014
[5] T. Sickenberger, R. Winkler, “Stochastic Oscillations in Circuit Simulation”, Proceeding in Applied Mathematics and Mechanics, 7:1 (2007), 4050023–4050024
[6] R. Winkler, “Stochastic DAEs in Transient Noise Simulation”, Proceedings of Scientific Computing in Electrical Engineering, 4 (2004), 408–415 · Zbl 1064.78008
[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), 4–9
[8] L.A. Vlasenko, S.L. Lyshko, A.G. Rutkas, “On a Stochastic Impulsive Sustem”, Reports of the National Academy of Sciences of Ukraine, 2012, no. 2, 50–55 · Zbl 1249.91084
[9] Belov A. A., Kurdyukov A. P., Descriptor Systems and Control Problems, Fizmatlit, M., 2015 (in Russian) · Zbl 1391.93001
[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, 148–158
[11] E.Yu. Mashkov, “Singular Stochastic Leontieff Type Equation with Depending on Time Diffusion Coefficients”, Global and Stochastic Analysis, 4:2 (2017), 207–217
[12] E. Nelson, “Derivation of the Schrödinger Equation from Newtonian Mechanics”, Physics Reviews, 150:4 (1966), 1079–1085
[13] E. Nelson, Dynamical Theory of Brownian Motion, Princeton University Press, Princeton, 1967 · Zbl 0165.58502
[14] E. Nelson, Quantum Fluctuations, Princeton University Press, Princeton, 1985 · Zbl 0563.60001
[15] Gliklikh Yu.E., Global and Stochastic Analysis in Mathematical Problems Physics, Comkniga, M., 2005
[16] Yu.E. Gliklikh, E.Yu. Mashkov, “Stochastic Leontieff Type Equation with Non-Constant Coefficients”, Applicable Analysis: An International Journal, 94:8 (2015), 1614–1623 · Zbl 1369.34079
[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, 6:2 (2013), 25–39 · Zbl 1307.60078
[18] Partasarati K. R., Introduction to Probability Theory and Measure Theory, Mir, M., 1988
[19] Gantmakher F. R., The Matrix Theory, Fizmatlit, M., 1967
[20] Gihman I. I., Scorohod A. V., Theory of Stochastic Processes, Springer, N.Y., 1979 · Zbl 0404.60061
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