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

Abstract

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.Under suitable conditions, the strong convergence orders are explicitly shown to be close to $\hat{I}\pm /2$, where $\hat{I}\pm \in (0,1)$ 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 $1/2$.Numerical simulations are provided to demonstrate the theoretical findings.