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

The importance of stochastic differential delay equations (SDDEs) derives from the fact that many of the phenomena witnessed around us do not have an immediate effect from the moment of their occurrence. A patient, for example, shows symptoms of an illness days (or even weeks) after the day in which he or she was infected. In general, we can find many ”systems”, in almost any area of science (medicine, physics, ecology, economics, etc.), for which the principle of causality, i.e., the future state of a system is independent of the past states and is determined solely by the present, does not apply. In order to incorporate this time lag (between the moment an action takes place and the moment its effect is observed) to our models, it is necessary to include an extra term which is called time delay. The SDDEs can be regarded as a generalization of stochastic differential equations (SDEs) and delay differential equations (DDEs). During the last few decades, many authors have studied SDDEs. some important results are given, for example, conditions which guarantee the existence and uniqueness of an analytical solution [13, 14, 15] and stability conditions for both exact solutions and numerical solutions, etc. [2, 6, 11, 16]. It is well known that in the deterministic situation there is a very special delay differential equation: the pantograph equation