  We study a time-inconsistent singular control problem originating from
irreversible reinsurance decisions with non-exponential discount. A novel
definition of equilibrium for time-inconsistent singular control problems is
introduced. For the problem with non-exponential discount, both sufficient and
necessary conditions are derived, providing a thorough mathematical
characterization of the equilibrium. Specifically, the equilibrium can be
characterized by an extended HJB system, which is a coupled system of non-local
parabolic equations with free boundaries. Finally, by showing the existence of
classical solutions to the extended HJB system, the existence of equilibria is
established under some additional assumptions.
