Equivalence of the mean square stability between the partially truncated Euler-Maruyama method and stochastic differential equations with super-linear growing coefficients

For stochastic differential equations (SDEs) whose drift and diffusion coefficients can grow super-linearly, the equivalence of the asymptotic mean square stability between the underlying SDEs and the partially truncated Euler-Maruyama method is studied. Using the finite time convergence as a bridge, a twofold result is proved. More precisely, the mean square stability of the SDEs implies that of the partially truncated Euler-Maruyama method, and the mean square stability of the partially truncated Euler-Maruyama method indicates that of the SDEs given the step size is carefully chosen.