  The semi-implicit Euler-Maruyama (EM) method is investigated to approximate a
class of time-changed stochastic differential equations, whose drift
coefficient can grow super-linearly and diffusion coefficient obeys the global
Lipschitz condition. The strong convergence of the semi-implicit EM is proved
and the convergence rate is discussed. When the Bernstein function of the
inverse subordinator (time-change) is regularly varying at zero, we establish
the mean square polynomial stability of the underlying equations. In addition,
the numerical method is proved to be able to preserve such an asymptotic
property. Numerical simulations are presented to demonstrate the theoretical
results.

Numerical Analysis;Numerical Analysis;Probability