Stability of stochastic differential equation driven by time-changed L\'evy noise

TL;DR

This paper investigates the stability properties of stochastic differential equations driven by time-changed Le9vy noise, establishing conditions for stability and exploring the relationship with original SDEs, with implications for various scientific fields.

Contribution

It provides new necessary conditions for stability of time-changed SDEs driven by Le9vy noise and links their stability to the original SDEs, extending previous results.

Findings

Established conditions for stability in probability and moments.

Connected stability of time-changed SDEs to original SDEs.

Provided examples illustrating different stability types.

Abstract

This paper studies stabilities of stochastic differential equation (SDE) driven by time-changed L\'evy 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 solution of time-changed SDE to be stable in different senses will be established. Connection between stability of 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 L\'evy noise, where the time-change processes are inverse of general L\'evy subordinators. These results are important improvements of the results in "Q. Wu, Stability of stochastic differential equation with respect to time-changed Brownian motion, 2016.".