Stochastic 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 L\'evy noises, which are in general not Markovian. To exploit the nature of the noise, we make use of different kind of information flows within a maximum principle approach. For this we work with backward stochastic differential equations (BSDE) with time-change and exploit the non-anticipating stochastic derivative as introduced in [7]. We prove both a stochastic sufficient and necessary maximum principle and we complete the work providing applications to optimal portfolio problems.