Volume 12, no. 1Pages 55 - 65

# Singular Stochastic Leontieff Type Equation in Current Velocities of Solutions

E.Yu. Mashkov, D.N. Tyutyunov
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.
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Keywords
mean derivative; current velocity; Wiener process; stochastic Leontieff type equation.
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