Precise Estimates for the Solution of Stochastic Functional Differential Equations with Discontinuous Initial Data: A Mathematical Approach

Abstract

In this chapter, we have proved a theorem in which we have established a uniform error bound for the Euler approximation to the solution process of the Stochastic Funtional Differential Equation (S.F.D.E.) (1.11) over the whole time interval . We have calculated this uniform error bound by computing the difference between the actual solution process and it’s Euler approximation and we have revealed the upper bound for this difference. This difference's reliance on the initial data has also been examined. We have also proved that the Euler approximation of the solution process has the order of strong convergence .\(\gamma=0.5\)