The $\alpha$-dependence of stochastic differential equations driven by variants of $\alpha$-stable processes

In this paper we investigate two variants of $\alpha$-stable processes, namely tempered stable subordinators and modified tempered stable process as well as their renormalization. We study the weak convergence in the Skorohod space and prove that they satisfy the uniform tightness condition. Finally, applications to the $\alpha$-dependence of the solutions of SDEs driven by these processes are discussed.