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In this paper, we consider stochastic differential equations with locally Lipschitz coefficients. Two classes of balanced Euler method are presented. Their moment boundedness and their convergence are considered. Their convergence rates are shown to be that of the classical Euler method. Moreover, the methods are showed that they can reproduce the mean square exponential stability of SDEs.

Convergence and stability of the two classes of balanced Euler methods for stochastic differential equations with locally Lipschitz coefficients