Stochastic Leontief Type Equations with Impulse Actions

Abstract

By a stochastic Leontief type equation we mean a special class of stochastic dierential equations in the Ito form, in which there is a degenerate constant linear operator in the left-hand side and a non-degenerate constant linear operator in the right-hand side. In addition, in the right-hand side there is a deterministic term that depends only on time, as well as impulse eects. It is assumed that the diusion coecient of this system is given by a square matrix, which depends only on time. To study the equations under consideration, it is required to consider derivatives of suciently high orders from the free terms, including the Wiener process. In connection with this, to dierentiate the Wiener process, we apply the machinery of Nelson mean derivatives of random processes, which makes it possible to avoid using the theory of generalized functions to the study of equations. As a result, analytical formulas are obtained for solving the equation in terms of mean derivatives of random processes.