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.
View Book:- https://stm.bookpi.org/RAMRCS-V9/article/view/6021
