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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.

advances in difference equations

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