  A linear-quadratic optimal control problem for a forward stochastic Volterra
integral equation (FSVIE, for short) is considered. Under the usual convexity
conditions, open-loop optimal control exists, which can be characterized by the
optimality system, a coupled system of an FSVIE and a Type-II backward SVIE
(BSVIE, for short). To obtain a causal state feedback representation for the
open-loop optimal control, a path-dependent Riccati equation for an
operator-valued function is introduced, via which the optimality system can be
decoupled. In the process of decoupling, a Type-III BSVIE is introduced whose
adapted solution can be used to represent the adapted M-solution of the
corresponding Type-II BSVIE. Under certain conditions, it is proved that the
path-dependent Riccati equation admits a unique solution, which means that the
decoupling field for the optimality system is found. Therefore a causal state
feedback representation of the open-loop optimal control is constructed. An
additional interesting finding is that when the control only appears in the
diffusion term, not in the drift term of the state system, the causal state
feedback reduces to a Markovian state feedback.
