ON THE STOCHASTIC SYSTEMS OF DIFFERENTIAL-ALGEBRAIC TYPE

Abstract

Under the stochastic system of differential-algebraic type we understand the special class of stochastic differential equations in the Ito form, in which in the left- and right-hand sides there are time-dependent continuous rectangular real matrices of the same size, and, in the case of a square matrix, the matrix in the left-hand side is degenerated. In addition, in the right-hand side there is a term that depends only on time. This class of equations is a natural generalization of the class of ordinary differential-algebraic equations. It is assumed that the initial conditions for this class of equations are solutions of some systems of linear algebraic equations, matrices in which are constant and have the same size as in the stochastic system. For the study of this class of equations, we use the machinery that is a generalization of the methods suggested for the study of ordinary differential-algebraic equations in the works by Yu. E. Boyarintsev, V. F. Chistyakov and others. Note that for investigation of these equations we do not use derivative of the right-hand side. We give the necessary information from the theory of pseudo-inverse matrix, and then thransform the system to a form more convenient for study. The result of the article is the statemants, in which sufficient conditions for the existence of solutions are obtained and formulae for finding the solutions are given.