In this paper, we consider stochastic differential equations whose drift coefficient is superlinearly growing and piece-wise continuous, and whose diffusion coefficient is superlinearly growing and locally H\"older continuous. We first prove the existence and uniqueness of the solution to such stochastic differential equations and then propose a tamed-adaptive Euler-Maruyama approximation scheme. We study the rate of convergence in the $L^1$-norm of the scheme in both finite and infinite time intervals.

翻译：本文考虑漂移系数为超线性增长且分段连续、扩散系数为超线性增长且局部Hölder连续的随机微分方程。我们首先证明此类随机微分方程解的存在唯一性，随后提出一种驯服自适应Euler-Maruyama逼近格式。我们研究了该格式在有限时间区间与无限时间区间上$L^1$范数的收敛速率。