On a Certain Approach to Investigation of Stochastic Differential Leontieff Type Equations

Abstract

In a finite-dimensional space, we consider a linear stochastic differential equation in Ito form, which has a degenerate constant matrix on the left side. Taking into account the various economic applications of these equations, they are classified as Leontief-type equations, since under some additional assumptions a deterministic analogue of the equation in question describes the famous input-output balance model of V. Leontief taking into account reserves.In the literature, these systems are more often called algebraic-differential and descriptor systems. In general, to study this type of equations it is necessary to consider higher-order derivatives of the right-hand side. This means that it is necessary to consider derivatives of the Wiener process that exist in a generalized sense. In previous works, these equations were studied using the apparatus of Nelson average derivatives of random processes, for the description of which generalized functions are not used. It is known that derivatives on average depend on which sigma-algebra is used to find them. In this work, the study of this equation was carried out using derivatives on average with respect to a new sigma-algebra, which was not considered in previous works.