Stability of the solution of stochastic differential equation driven by time-changed Lévy noise

Abstract

This paper studies stabilities of the solution of stochastic differential equations (SDE) driven by time-changed Lévy noise in both probability and moment sense. This provides more flexibility in modeling schemes in application areas including physics, biology, engineering, finance and hydrology. Necessary conditions for the solution of time-changed SDE to be stable in different senses will be established. The connection between stability of the solution to time-changed SDE and that to corresponding original SDE will be disclosed. Examples related to different stabilities will be given. We study SDEs with time-changed Lévy noise, where the time-change processes are the inverse of general Lévy subordinators. These results are an important generalization of the results of Q. Wu (2016).