Our subject of study is strong approximation of stochastic differential equations (SDEs) with respect to the supremum error criterion, and we seek approximations that are strongly asymptotically optimal in specific classes of approximations. We hereby focus on two principal types of classes, namely, the classes of approximations that are based only on the evaluation of the initial value and on at most finitely many sequential evaluations of the driving Brownian motion on average, and the classes of approximations that are based only on the evaluation of the initial value and on finitely many evaluations of the driving Brownian motion at equidistant sites. For SDEs with globally Lipschitz continuous coefficients, Müller-Gronbach [Ann. Appl. Probab. 12 (2002), no. 2, 664–690] showed that specific Euler–Maruyama schemes relating to adaptive and to equidistant time discretizations are strongly asymptotically optimal in these classes. In the present article, we generalize the results above to a significantly wider class of SDEs, such as ones with super-linearly growing coefficients. More precisely, we prove strong asymptotic optimality for specific coefficient-modified Euler–Maruyama schemes relating to adaptive and to equidistant time discretizations under rather mild assumptions on the underlying SDE. To illustrate our findings, we present two exemplary applications – namely, Euler–Maruyama schemes and tamed Euler schemes – and thereby analyze the SDE associated with the Heston-$3∕2$-model originating from mathematical finance.

我们的研究主题是相对于最高误差准则的随机微分方程（SDE）的强逼近，并且我们寻求在特定逼近类中强烈渐近最优的逼近。在此，我们关注两种主要类型的类别，即仅基于初始值的评估和平均有限的驱动布朗运动的最多连续有限评估的近似类别，以及仅基于初始布朗运动的近似有限类。仅基于对初始值的评估以及对等距站点上驱动布朗运动的有限评估。对于具有全局Lipschitz连续系数的SDE，Müller-Gronbach[Ann。应用 Probab。12（2002），no。2，[664-690]表明，在这些类别中，与自适应和等距时间离散化有关的特定欧拉-丸山方案在渐近最优。在本文中，我们将上述结果推广到更广泛的SDE类，例如具有超线性增长系数的SDE。更准确地说，我们证明了在基本SDE的相当温和的假设下，与自适应和等距时间离散化有关的特定系数修改的Euler-Maruyama方案的强渐近最优性。为了说明我们的发现，我们提出了两个示例性应用程序，即Euler-Maruyama方案和驯服的Euler方案，从而分析了与Heston- 我们将上述结果推广到更广泛的SDE类，例如具有超线性增长系数的SDE。更准确地说，我们证明了在基本SDE的相当温和的假设下，与自适应和等距时间离散化有关的特定系数修改的Euler-Maruyama方案的强渐近最优性。为了说明我们的发现，我们提出了两个示例性应用程序，即Euler-Maruyama方案和驯服的Euler方案，从而分析了与Heston- 我们将上述结果推广到更广泛的SDE类，例如具有超线性增长系数的SDE。更准确地说，我们证明了在基本SDE的相当温和的假设下，与自适应和等距时间离散化有关的特定系数修改的Euler-Maruyama方案的强渐近最优性。为了说明我们的发现，我们提出了两个示例性应用程序，即Euler-Maruyama方案和驯服的Euler方案，从而分析了与Heston-$3∕2$源自数学金融的模型。