Well-posedness and an Euler-Maruyama method for multi-term caputo tempered fractional stochastic differential equations

Abstract

In this paper, we first prove the existence, uniqueness and continuity dependence on the initial value of multi-term Caputo tempered fractional stochastic differential equations with initial value. Then, an Euler-Maruyama (EM) method is presented for solving the considered equation. The strong convergence of the presented EM method is strictly investigated with the order to be $\min\{\alpha_m-\alpha_{m-1},\alpha_{m}-0.5\}$ with $0<\alpha_{1}<\alpha_{2}<\cdots<\alpha_{m-1}<\alpha_{m}<1$, $\alpha_{m}>0.5$, and $m$ is a given positive integer. Finally, three numerical examples are provided to support our theoretical findings.