  In this paper, we study extended backward stochastic Volterra integral
equations (EBSVIEs, for short). We establish the well-posedness under weaker
assumptions than the literature, and prove a new kind of regularity property
for the solutions. As an application, we investigate, in the open-loop
framework, a time-inconsistent stochastic recursive control problem where the
cost functional is defined by the solution to a backward stochastic Volterra
integral equation (BSVIE, for short). We show that the corresponding adjoint
equations become EBSVIEs, and provide a necessary and sufficient condition for
an open-loop equilibrium control via variational methods.
