  In this paper, we prove both necessary and sufficient maximum principles for
infinite horizon discounted control problems of stochastic Volterra integral
equations with finite delay and a convex control domain. The corresponding
adjoint equation is a novel class of infinite horizon anticipated backward
stochastic Volterra integral equations. Our results can be applied to
discounted control problems of stochastic delay differential equations and
fractional stochastic delay differential equations. As an example, we consider
a stochastic linear-quadratic regulator problem for a delayed fractional
system. Based on the maximum principle, we prove the existence and uniqueness
of the optimal control for this concrete example and obtain a new type of
explicit Gaussian state-feedback representation formula for the optimal
control.
