Euler–Maruyama methods for Caputo tempered fractional stochastic differential equations

Abstract

In this paper, we introduce the initial value problem of Caputo tempered fractional stochastic differential equations and then study the well-posedness of its solution. Further, a Euler–Maruyama (EM) method is derived for solving the considered problem. The strong convergence order of the derived EM method is proved to be $\alpha -\frac{1}{2}$ with $\frac{1}{2}<\alpha <1$. Additionally, a fast EM method is also developed which is based on the sum-of-exponentials approximation. Finally, numerical experiments are given to support the theoretical findings of the above two methods and verify computational efficiency of the fast EM method. The fast EM method can greatly improve the computational performance of the original EM method.

KEYWORDS:

• Tempered fractional stochastic differential equations
• well-posedness
• Euler–Maruyama method
• convergence
• fast computation

MATHEMATICS SUBJECT CLASSIFICATIONS (2010):

• 65C30
• 60H10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is supported by Natural Science Foundation of Jiangsu Province of China (Grant No. BK20201427), and by National Natural Science Foundation of China (Grant Nos. 11701502 and 11871065).