A note on strong convergence of implicit scheme for SDEs under local one-sided Lipschitz conditions

Under a local one-sided Lipschitz condition, Krylov [KR] proved the existence and uniqueness of the strong solutions for stochastic differential equations by using the Euler-Maruyama approximation, where he showed that the sequence of numerical solutions converges to the true solution in probability as the stepsize tends to zero. In this note, we shall extend the results in [KR] and investigate an implicit numerical scheme for these equations under a local one-sided Lipschitz condition.