Consistency and Numerical Stability of the Euler Approximation for the Solution of Delay SDE’s with Possibly Discontinuous Initial Data
This study follows on from ’s work on Precise Estimates and ’s work on Approximation Theorems. We have shown that the S.F.D.E. Euler approximation considered in  and  is numerically stable and weakly consistent. Note that the introduction, notations, and definitions are the same as in  and . We estimated the uniform error bound of the difference between the real solution process and its Euler approximation and determined the upper bound for this difference in , , , and . We’ve also spoken about how the uniform error is affected by the original data. We also estimated the inaccuracy in this work by taking the original value into account. The beginning data of the solution process is equivalent to the data of the Euler approximation. We also estimated the inaccuracy in this study if the original initial data was substituted with another initial data. We also analysed the differences in the solution procedures acquired from various starting data.
Tagelsir A. Ahmed
Department of Pure Mathematics, Faculty of Mathematical Science, University of Khartoum, P.O.Box 321, Khartoum, Sudan.
A. Van Casteren, Jan
Department of Mathematics and Computer Science, University of Antwerp (UA), Middelheimlaan 1, 2020 Antwerp, Belgium.
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