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This paper studies stabilities of stochastic differential equation (SDE)driven by time-changed L\'evy noise in both probability and moment sense. Thisprovides more flexibility in modeling schemes in application areas includingphysics, biology, engineering, finance and hydrology. Necessary conditions forsolution of time-changed SDE to be stable in different senses will beestablished. Connection between stability of solution to time-changed SDE andthat to corresponding original SDE will be disclosed. Examples related todifferent stabilities will be given. We study SDEs with time-changed L\'evynoise, where the time-change processes are inverse of general L\'evysubordinators. These results are important improvements of the results in "Q.Wu, Stability of stochastic differential equation with respect to time-changedBrownian motion, 2016.".

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