  For the stochastic differential equation (SDE) which has piecewise continuous
arguments (PCAs), is driven by multiplicative noises and its drift coefficients
are dissipative, we show that the solution at integer time is a Markov chain
and admits a unique invariant measure. In order to inherit numerically the
invariant measure of SDE with PCAs, we apply the backward Euler (BE) method to
the equation, and prove that the numerical solution at integer time is not only
Markovian but also reproduces a unique numerical invariant measure. We present
the time-independent weak error analysis for the method under certain
hypothesis. Further, we show that the numerical invariant measure converges to
the original one with order 1. Numerical experiments verify the theoretical
analysis.
