Singular Stochastic Leontieff Type Equation in Current Velocities of Solutions

We investigate the system of stochastic differential equations, such that in the left-hand and right-hand sides there are rectangular constant matrices forming degenerate pencil. The system is considered in terms of current velocities of solution that are a direct analogue of physical velocity of deterministic processes. For investigation of this system we apply the Kronecker-Weierstrass transformation of the pencil of matrices coefficients to the canonical form that efficiently simplifies the investigation. As a result, the canonical system splits into independent sub-systems of four types. For the sub-systems corresponding to the Jordan singular Kronecker's cells, we obtain the explicit formulae of solutions and conditions for solvability. For the sub-system resolved with respect to symmetric derivatives, we apply the replacement of the metric in the subspace, then bring the system to a stochastic equation in the Ito form and prove the existence of its solution. As a result for the system under consideration we prove the existence of the solution theorem under some additional conditions on the coefficients.

Keywords

mean derivative; current velocity; Wiener process; stochastic Leontieff type equation.

References

1. Zamyshlyaeva A.A., Keller A.V., Syropiatov M.B. Stochastic Model of Optimal Dynamic Measurements. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2018, vol. 11, no. 2, pp. 147-156. DOI: 10.14529/mmp180212
2. Shestakov A.L., Sviridyk G.A., Butakova M.D. The Mathematical Modelling of the Production of Construction Mixtures with Prescribed Properties. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 1, pp. 100-110. DOI: 10.14529/mmp150108
3. Belov A.A., Kurdyukov A.P. Deskriptornye sistemy i zadachi upravleniya [Descriptor Systems and Control Problems]. Moscow, Fizmatlit, 2015. (in Russian)
4. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics, London, Springer, 2011. DOI: 10.1007/978-0-85729-163-9
5. Gliklikh Yu.E., Mashkov E.Yu. Stochastic Leontieff Type Equations in Terms of Current Velocities of the Solution. Journal of Computational and Engineering Mathematics, 2014, vol. 1, no. 2, pp. 45-51.
6. Gliklikh Yu.E., Mashkov E.Yu. Stochastic Leontieff Type Equations in Terms of Current Velocities of the Solution II. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2016, vol. 9, no. 3, pp. 31-40. DOI: 10.14529/mmp160303
7. Mashkov E.Yu. On the Stochastic Leontief Type Equations with Variable Matrices Given in Terms of Current Velocities of the Solution. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2016, vol. 8, no. 4, pp. 26-32. DOI: 10.14529/mmph160403
8. Nelson E. Dynamical Theory of Brownian Motion. Princeton, Princeton University Press, 1967.
9. Nelson E. Quantum Fluctuations. Princeton, Princeton University Press, 1985.
10. Parthasaraty K.R. Introduction to Probability and Measure. N.Y., Springer, 1978.
11. Gihman I.I. Theory of Stochastic Processes. N.Y., Springer, 1979. DOI: 10.1007/978-1-4615-8065-2
12. Demmel J. Applied Numerical Linear Algebra. Philadelphia, Society for Industrial and Applied Mathematics, 1997. DOI: 10.1137/1.9781611971446