  In this paper, we consider a fixed delay Cox-Ingersoll-Ross process (CIR
process) on the regime where it does not hit zero, the aim is to determine a
positive preserving implicit Euler Scheme. On a time grid with constant
stepsize our scheme extends the scheme proposed by Alfonsi in 2005 for the
classical CIR model. Furthermore, we consider its piecewise linear
interpolation, and, under suitable conditions, we establish the order of strong
convergence in the uniform norm, thus extending the results of Dereich et al.
in 2011.
