Uniform Convergence of Euler Approximation of the Solution of Stochastic Functional Differential Equations with Discontinuous Initial Data

Since the model Delay SDE's are generally non-linear and do not allow for explicit solutions, there is a clear need for numerical approximation methods for solutions of delay stochastic equations. Early investigations in this direction were investigated in [1] and [2]. Many models of physical phenomena happen to be stochastic equations. For many of these stochastic differential equations, we can not find the solution explicitly, but we can use suitable approximation method to get an approximate solution for our ordinary SDE. Here "Stochastic Functional Differential Equations (S.F.D.E’s)" means ’Delay Stochastic Differential Equations". In this work we have developed an Euler approximation scheme for the solution process of Stochastic Functional Differential Equation with possibly discontinuous initial data, and we have shown that this Euler scheme (under appropriate conditions) converges to the solution process as the mesh of the partition goes to zero, see also[3] and [4].

The approximation theorem which we have established gives us a method for approximating the solution of S.F.D.E’s with possibly discontinuous initial data. Note that here we are considering S.F.D.E which includes both drift and diffusion coefficients. The present work on approximation is an extension of the work on approximation in [1] to include S.F.D.E’s with both drift and diffusion coefficients. The work on approximation in [1] was suggested by Prof. Salah-E. A. Mohammed and it was done by Tagelsir A. Ahmed under the supervision of Prof. Salah-E. A. Mohammed.