This paper introduces a randomized tamed Euler scheme tailored for L\'evy-driven stochastic differential equations (SDEs) with superlinear random coefficients and Carath\'eodory-type drift. Under assumptions that allow for time-irregular drifts while ensuring appropriate time-regularity of the diffusion and jump coefficients, the proposed scheme is shown to achieve the optimal strong $\mathcal{L}^2$-convergence rate, arbitrarily close to $0.5$. A crucial component of our methodology is the incorporation of drift randomization, which overcomes challenges due to low time-regularity, along with a taming technique to handle the superlinear state dependence. Our analysis moreover covers settings where the coefficients are random, providing for instance strong convergence of randomized tamed Euler schemes for L\'evy-driven stochastic delay differential equations (SDDEs) with Markovian switching. To our knowledge, this is the first {work} that addresses the case of superlinear coefficients in the numerical analysis of Carath\'eodory-type SDEs and even for ordinary differential equations.

翻译：本文提出了一种针对具有超线性随机系数和Carathéodory型漂移项的Lévy驱动随机微分方程(SDEs)的随机化驯服欧拉格式。在允许时间不规则漂移项、同时确保扩散项和跳跃系数具有适当时间正则性的假设下，所提格式被证明能达到最优强$\mathcal{L}^2$收敛阶，且可任意接近0.5。我们方法的一个关键组成部分是引入了漂移项随机化，以克服低时间正则性带来的挑战，并结合驯服技术来处理状态依赖的超线性增长。此外，我们的分析还涵盖了系数为随机的情形，例如为具有马尔可夫切换的Lévy驱动随机延迟微分方程(SDDEs)的随机化驯服欧拉格式提供了强收敛性证明。据我们所知，这是首个在Carathéodory型SDEs乃至常微分方程的数值分析中处理超线性系数情形的研究工作。