Convergence and Stability of the Semi-Implicit Euler Method with Variable Stepsize for a Linear Stochastic Pantograph Differential Equation

Abstract

The paper deals with convergence and stability of the semi-implicit Euler method with variable stepsize for a linear stochastic pantograph differential equation (SPDE). It is proved that the semi-implicit Euler method with variable stepsize is convergent with strong order $p = \frac{1}{2}$. The conditions under which the method is mean square stability are determined and the numerical experiments are given.