Backward Volterra equations with time-changed L\'evy noise and maximum principles

We study an optimal control problem for Volterra type dynamics driven by time-changed Levy noises with a maximum principle approach. For this we use different kind of information flows, the non anticipating stochastic derivative and we study backward Volterra integral equations (BSVIE) with time-change. We also provide an explicit solution for the linear BSVIEs. With these, we prove both a stochastic Pontryagin and Mangasarian maximum principle. We complete the work providing applications to optimal portfolio problems, particularly mean variance selection.