Khasminskii-type theorems for stochastic functional differential equations

Abstract

For a stochastic functional differential equation (SFDE) to have aunique global solution it is in general required that thecoefficients of the SFDE obey the local Lipschitz condition and thelinear growth condition. However, there are many SFDEs in practicewhich do not obey the linear growth condition. The main aim of thispaper is to establish existence-and-uniqueness theorems for SFDEswhere the linear growth condition is replaced by more generalKhasminskii-type conditions in terms of a pair of Laypunov-typefunctions.