Stability of solutions of stochastic functional-differential equations with poisson switchings and entire prehistory

Abstract

The Markov properties of the solutions of Ito-Skorokhod stochastic functional-differential equations (SFDEs) with entire prehistory are considered, the concept of a weak infinitesimal operator is introduced for a Markov process that is a solution of an SFDE, and the strong solution of the SFDE is analyzed for stability.The concept of Lyapunov stability, namely, the method of Lyapunov functions (Lyapunov-Krasovskii functionals)[10] is the most constructive and universal approach to the stability analysis of deterministic systems. The main advantage of this method is that it allows concluding whether a system whose analytical solution cannot be found by the direct integration of the differential equations is stable.In developing the theory of stability of stochastic systems, the following aspects are fundamentally important: · considering systems that possess the Markov property; · the concept of strong probabilistic stability, which makes it possible to analyze the asymptotic behavior of realizations of a random solution-process of the corresponding stochastic equation; · using the method of functions (functionals), namely, the concept of a derivative for deterministic systems or a weak infinitesimal operator for stochastic dynamic systems which can be computed without the need to integrate the equation of perturbed motion.