This paper deals with a family of balanced methods which own the implicit iterative scheme in the diffusion term for the stochastic delay integro-differential equations. It is shown that the balanced implicit methods, which are fully implicit methods, give strong convergence rate of at least 1/2 and that the strong balanced methods can preserve the linear mean-square stability with the sufficiently small stepsize. Weak variants are also considered and their mean-square stability is analyzed. Some numerical experiments are given to demonstrate the conclusions and to show that the fully implicit methods are superior to those of the explicit methods in terms of mean-square stabilities.

stochastic delay integro-differential equation, balanced method, convergence, mean-square stability.