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

TL;DR

This paper explores how solutions to stochastic differential equations driven by tempered stable processes depend on the stability parameter alpha, analyzing convergence and tightness properties.

Contribution

It introduces and studies two variants of alpha-stable processes, providing new insights into their convergence and applications to SDEs.

Findings

Proved weak convergence of tempered stable variants in Skorohod space

Established uniform tightness conditions for these processes

Discussed implications for alpha-dependent SDE solutions

Abstract

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.