In this paper, we formulate and investigate the notion of causal feedback
strategies arising in linear-quadratic control problems for stochastic Volterra
integral equations (SVIEs) with singular and non-convolution-type coefficients.
We show that there exists a unique solution, which we call the causal feedback
solution, to the closed-loop system of a controlled SVIE associated with a
causal feedback strategy. Furthermore, introducing two novel equations named a
Type-II extended backward stochastic Volterra integral equation and a
Lyapunov--Volterra equation, we prove a duality principle and a representation
formula for a quadratic functional of controlled SVIEs in the framework of
causal feedback strategies.Comment: 29 page