We study the convergence rates of the semi-discrete (SD) method originally proposed in Halidias (2012), Semi-discrete approximations for stochastic differential equations and applications, International Journal of Computer Mathematics, 89(6). The SD numerical method was originally designed mainly to reproduce qualitative properties of nonlinear stochastic differential equations (SDEs). The strong convergence property of the SD method has been proved, but except for certain classes of SDEs, the order of the method was not studied. We study the order of L2-convergence and show that it can be arbitrarily close to 1/2. The theoretical findings are supported by numerical experiments.

中文翻译：

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随机微分方程的半离散方法的收敛率

我们研究了最初在 Halidias (2012)，随机微分方程和应用的半离散近似，国际计算机数学杂志，89(6) 中提出的半离散 (SD) 方法的收敛率。SD 数值方法最初的设计主要是为了重现非线性随机微分方程 (SDE) 的定性特性。已经证明了 SD 方法的强收敛性，但除某些类别的 SDE 外，没有研究该方法的阶数。我们研究了 L2 收敛的阶数，并表明它可以任意接近 1/2。理论发现得到了数值实验的支持。