A class of stochastic one-parameter methods for nonlinear SFDEs with piecewise continuous arguments

Abstract

This paper deals with nonlinear stochastic functional differential equations with piecewise continuous arguments (SFDEPCAs). Based on an adaptation to the underlying one-leg θ-methods for ODEs, a class of new one-parameter methods for nonlinear SFDEPCAs are introduced. The mean-square exponential stability criteria of analytical and numerical solutions are derived. Under the suitable conditions, it is proved that the one-parameter methods are convergent with strong order 1/2. Some numerical experiments are given to illustrate the theoretical results and computational advantages of the induced methods.