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.

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关于局部单边 Lipschitz 条件下 SDE 隐式方案的强收敛性的注记

在局部单边 Lipschitz 条件下，Krylov [KR] 通过使用 Euler-Maruyama 近似证明了随机微分方程强解的存在性和唯一性，他证明了数值解的序列在概率上收敛到真解因为步长趋于零。在本笔记中，我们将扩展 [KR] 中的结果，并在局部单边 Lipschitz 条件下研究这些方程的隐式数值方案。