  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.".
