An Approach to Studying Leontief Type Stochastic Differential Equations

Abstract

In a finite-dimensional space, we consider a linear stochastic differential equation in Itô form with a singular constant matrix on the left-hand side. Taking into account various economic applications of such equations, they are classified as Leontief type equations, since under some additional assumptions, a deterministic analog of the equation in question describes the famous Leontief input–output balance model taking into account reserves. In the literature, these systems are more often called differential–algebraic or descriptor systems. In general, to study this type of equations, one needs higher-order derivatives of the right-hand side. This means that one must consider derivatives of the Wiener process, which exist in the generalized sense. In the previous papers, these equations were studied using the technique of Nelson mean derivatives of random processes, whose description does not require generalized functions. It is well known that mean derivatives depend on the \(\sigma\)-algebra used to find them. In the present paper, the study of this equation is carried out using mean derivatives with respect to a new \(\sigma\)-algebra that was not considered in the previous papers.