  We study backward stochastic differential equations (BSDEs) for time-changed
L\'evy noises when the time-change is independent of the L\'evy process. We
prove existence and uniqueness of the solution and we obtain an explicit
formula for linear BSDEs and a comparison principle. BSDEs naturally appear in
control problems. Here we prove a sufficient maximum principle for a general
optimal control problem of a system driven by a time-changed L\'evy noise. As
an illustration we solve the mean-variance portfolio selection problem.
