Singular stochastic leontieff type equation in current velocities of solutions

E. Yu. Mashkov ; D. N. Tyutyunov

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 12 (2019) no. 1, pp. 55-65

Résumé

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.

Détail

Keywords: mean derivative, current velocity, Wiener process, stochastic Leontieff type equation.