Uniform Convergence of Euler Approximation of the Solution of Stochastic Functional Differential Equations with Discontinuous Initial Data | Chapter 03 | Recent Recent Advances in Mathematical Research and Computer Science Vol. 9

 Because model Delay SDEs are often non-linear and do not allow for explicit solutions, numerical approximation approaches for solutions of delay stochastic equations are clearly required. Early explorations in this area were conducted in [1] and [2]. Many physical phenomenon models are stochastic equations. Many of these stochastic differential equations do not have an explicit solution, but we can apply an appropriate approximation approach to obtain an approximate solution for our ordinary SDE. "Stochastic Functional Differential Equations (S.F.D.E's)" in this context refers to "Delay Stochastic Differential Equations." We developed an Euler approximation scheme for the solution process of a Stochastic Functional Differential Equation with possibly discontinuous initial data in this work, and we demonstrated that this Euler scheme (under appropriate conditions) converges to the solution process as the mesh of the partition approaches zero, see also[3] and [4].

The approximation theorem we proposed provides a mechanism for approximating the solution of S.F.D.E's with potentially discontinuous initial data. It's worth noting that we're looking at S.F.D.E. here, which contains both drift and diffusion coefficients. The current approximation study extends the approximation work in [1] to include S.F.D.E's with both drift and diffusion coefficients. Prof. Salah-E. A. Mohammed recommended the work on approximation in [1], which was completed by Tagelsir A. Ahmed under Prof. Salah-E. A. Mohammed's supervision.

Author(S) Details

Tagelsir A. Ahmed
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Khartoum, Sudan.

J. A. Van Casteren
Department of Mathematics and Computer Science, University of Antwerp (UA), Middelheimlaan 1, 2020 Antwerp, Belgium.

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