Devoted to numerical solutions of stochastic differential equations (SDEs),
in this talk we construct a sequence of re-embedded numerical solutions having the
same distribution as that of the original SDE in a new probability space. It is shown
that the re-embedded numerical solutions converge strongly to the solution of the SDE.
Moreover, the rate of convergence is ascertained. Different from the well-known results
in numerical solutions of SDEs, in lieu of the usually used Brownian motion increments
in the algorithm, an easily implementable sequence of independent and identically
distributed (i.i.d.) random variables is used. Being easier to implement compared to the
construction of Brownian increments, such an i.i.d. sequence is preferable in the actual
computation. As far as the convergence and uniform mean squares error estimates are
concerned, the use of the i.i.d. sequence does not introduce essential difficulties compared
with that of the Brownian increments. Nevertheless, the analysis becomes much more
difficult for the study of rates of convergence because one has to deal with the diference
of the Brownian increments and the i.i.d. sequence in the almost sure sense.
Our work presents a new angle of ascertaining the convergence rates.