Expansion of the global error for numerical schemes solving stochastic differential equations

Given the solution (Xt ) of a Stochastic Differential System, two situat,ions are considered: computat,ion of Ef(Xt ) by a Monte–Carlo method and, in the ergodic case, integration of a function f w.r.t. the invariant probability law of (Xt ) by simulating a simple t,rajectory. For each case it is proved the expansion of the global approximat,ion error—for a class of discret,isat,ion schemes and of funct,ions f—in powers of the discretisation step size, extending in the fist case a result of Gragg for deterministic O.D.E. Some nn~nerical examples are shown to illust,rate the applicat,ion of extrapolation methods, justified by the foregoing expansion, in order to improve the approximation accuracy