Parameter-related strong convergence rates of Euler-type methods for time-changed stochastic differential equations

An Euler-type framework with equidistant step sizes is proposed for a class of time-changed stochastic differential equations.We establish the strong convergence rate of the standard Euler–Maruyama method under the global Lipschitz condition.The theoretical analysis is then extended to the truncated Euler–Maruyama method, proving its strong convergence under relaxed Khasminskii-type conditions.For both numerical schemes, the strong convergence orders are explicitly shown to be close to $\alpha \mathrm{/}2$, where $\alpha \in (0,1)$ is the parameter of the time-change process.These results are significantly different from existing works using random step sizes, which typically preserve the classical convergence order of $1\mathrm{/}2$.Numerical simulations are provided to demonstrate the theoretical findings.