Consistency and Numerical Stability of the Euler Approximation for the Solution of Delay SDE's with Possibly Discontinuous Initial Data

This work is a continuation to the work on Precise Estimates in [1] and the work on Approximation Theorems in [2]. Here we have proved that the Euler approximation of the S.F.D.E. considered in [2] and [1] is in fact numerically stable and weakly consistent. Note that here we have used the same introduction, notations and definitions as in [2] and [1]. In [2], [1], [3] and [4] we have calculated the uniform error Bound of the difference between the actual solution process and it's Euler approximation and we have found the upper bound for this difference. Here we have also discussed the dependence of the uniform error on the initial data. Also in this work we have calculated the error by considered the initial data of the Euler approximation equals to the initial data of the solution process. Also in this work we have calculated the error in case that the original initial data is replaced by different initial data. Also we have compared the difference of the solution processes obtained by different initial data.