In this paper, we consider a kind of multi-term Caputo tempered fractional stochastic differential equations and prove the existence and uniqueness of the true solution. Then we derive an Euler–Maruyama (EM) scheme to solve the considered equations. In view of the huge computational cost caused by the EM scheme to achieve reasonable accuracy, a fast EM scheme is proposed based on the sum-of-exponentials approximation to improve its computational efficiency. Moreover, the strong convergence of our two numerical schemes are proved. Finally, several numerical examples are carried out to support our theoretical results and demonstrate the superior computational efficiency of the fast EM scheme.

中文翻译：

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多项Caputo调质分数阶随机微分方程的快速Euler-Maruyama格式及其强收敛性


本文考虑一类多项Caputo调质分数阶随机微分方程，并证明真解的存在性和唯一性。然后我们推导出欧拉-丸山 (EM) 方案来求解所考虑的方程。鉴于EM方案要达到合理的精度所造成的巨大计算成本，提出了一种基于指数和近似的快速EM方案以提高其计算效率。此外，证明了我们的两个数值格式的强收敛性。最后，进行了几个数值例子来支持我们的理论结果，并证明了快速 EM 方案卓越的计算效率。