An adaptive weak continuous Euler-Maruyama method for stochastic delay differential equations

In this paper, an adaptive weak scheme for stochastic delay differential equations (SDDEs) based on the weak continuous Euler-Maruyama method which is a special member of the family of continuous weak Runge-Kutta schemes is introduced. The framework of the analysis of the global error is to embed the SDDE into a series of interrelated SDEs each defined on a separate interval in order to consider the error of SDE method and that of the interpolation. We perform the error estimation in a priori form based on the rooted tree theory of Roler and then analyze the global error of the scheme by obtaining a computable expression of the principal terms of that which is useful for controlling it which contains both the numerical and statistical errors. Adopting the idea presented in Szepessy et al. (Commun. Pure Appl. Math. 54:1169-1214, 2001), we determine the optimal discretization points using the deterministic time-step mechanism and also the necessary number of realizations based on the standard deviation of the approximate solution. We show that this technique leads to increased accuracy of the expected value of the required functionals. By presenting some numerical experiments, the effectiveness of utilizing the adaptive idea with holding the tolerance proportionality property is illustrated.