  An optimal control problem is considered for a stochastic differential
equation with the cost functional determined by a backward stochastic Volterra
integral equation (BSVIE, for short). This kind of cost functional can cover
the general discounting (including exponential and non-exponential) situation
with a recursive feature. It is known that such a problem is time-inconsistent
in general. Therefore, instead of finding a global optimal control, we look for
a time-consistent locally near optimal equilibrium strategy. With the idea of
multi-person differential games, a family of approximate equilibrium strategies
is constructed associated with partitions of the time intervals. By sending the
mesh size of the time interval partition to zero, an equilibrium
Hamilton--Jacobi--Bellman (HJB, for short) equation is derived, through which
the equilibrium valued function and an equilibrium strategy are obtained. Under
certain conditions, a verification theorem is proved and the well-posedness of
the equilibrium HJB is established. As a sort of Feynman-Kac formula for the
equilibrium HJB equation, a new class of BSVIEs (containing the diagonal value
$Z(r,r)$ of $Z(\cd,\cd)$) is naturally introduced and the well-posedness of
such kind of equations is briefly presented.
