Investigation of Leontieff Type Equations with White Noise protect by the Methods of Mean Derivatives of Stochastic Processes

Abstract

We understand the Leontieff type equation with white noise as the expression of the form $L\dot\xi(t)=M\xi(t)+\dot w(t)$ where $L$ is a degenerate matrix $n\times n$, $M$ is a non-degenerate matrix $n\times n$, $\xi(t)$ is a stochastic process we are looking for and $\dot w(t)$ is the white noise. Since the derivative $\dot\xi(t)$ and the white noise are well-posed only in terms of distributions, the direct investigation of such equations is very complicated. We involve two methods in the investigation. First, we pass to the stochastic differential equation $L\xi(t)=M\int_0^t\xi(s)ds+w(t)$, where $w(t)$ is Wiener process, and then for describing solutions of this equations we apply the so called Nelson mean derivatives that are introduced without using the distributions. By these methods we obtain formulae for solutions of Leotieff type equations with white noise.