Convergence and stability of the semi-implicit Euler method for linear stochastic delay integro-differential equations

Delay integro-differential equations are very important in biology, medicine and many other fields. If we take random noise into account, we can obtain many stochastic delay integro-differential equations (SDIDEs). As a special case of stochastic functional differential equations (SFDEs), the fundamental theory of existence and uniqueness of the solution of SDIDEs can be regarded similarly to the SFDES studied previously [1, 2]. An explicit solution can rarely be obtained for SDIDEs. It is necessary to develop numerical methods and to study the properties of these methods. Convergence and stability are the basic properties of numerical methods. There are many results for the numerical solutions of SOD equations [3–7], and various studies have been carried out on the stability of stochastic difference equations [8–10]. The study of the stability of the numerical solutions to SFDEs is in its infancy [11, 12], and there is little research on the numerical solutions to SDIDEs. Koto [13] studied the stability of the numerical methods for the test equation