A derivative-free explicit method with order 1.0 for solving stochastic delay differential equations

A new explicit stochastic scheme of order 1 is proposed for solving stochastic delay differential equations (SDDEs) with sufficiently smooth drift and diffusion coefficients and a scalar Wiener process. The method is derivative-free and is shown to be stable in mean square. A stability theorem for the continuous strong approximation of the solution of a linear test equation by the Milstein method is also proved, which shows the stepsize restriction for stability is larger than those given previously in the literature. The case of linear SDDEs is further investigated, in order to compare the stepsize restrictions for stability of these two methods. Numerical experiments are given to illustrate the obtained stability properties.