  In this paper numerical methods for solving stochastic differential equations
with Markovian switching (SDEwMSs) are developed by pathwise approximation. The
proposed family of strong predictor-corrector Euler-Maruyama methods is
designed to overcome the propagation of errors during the simulation of an
approximate path. This paper not only shows the strong convergence of the
numerical solution to the exact solution but also reveals the order of the
error under some conditions on the coefficient functions. A natural analogue of
$p$-stability criterion is studied. Numerical examples are given to illustrate
the computational efficiency of the new predictor-corrector Euler-Maruyama
approximation.
