Antoniuk S. Properties of solutions of stochastic differential-functional equations with infinite afteraction

The behavior of solution of Ito-Skorokhod stochastic dynamic systems with all prehistory is investigated in this thesis. Existence and uniqueness theorems, theorem of continuous dependence on initial conditions are proved for strong solution of such equations. Second Lyapunov method for investigation of stability of solutions of such stochastic dynamic systems is elaborated, the class of Lyapunov-Krasovsky functionals and methods of calculating the weak infinitesimal operator on solutions of stochastic functional-differential equations with all prehistory are described, theorems of weak stability a.s. are proved. Conditions of asymtotical mean square stability of solutin of linear Ito-Skorokhof stochastic dynamic systems are obtained. This theoretical results was applicated into analysis of stability of solution of stochasic problem “Dandling spider”.