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

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