  An optimal control problem is considered for a stochastic differential
equation containing a state-dependent regime switching, with a recursive cost
functional. Due to the non-exponential discounting in the cost functional, the
problem is time-inconsistent in general. Therefore, instead of finding a global
optimal control (which is not possible), we look for a time-consistent
(approximately) locally optimal equilibrium strategy. Such a strategy can be
represented through the solution to a system of partial differential equations,
called an equilibrium Hamilton-Jacob-Bellman (HJB, for short) equation which is
constructed via a sequence of multi-person differential games. A verification
theorem is proved and, under proper conditions, the well-posedness of the
equilibrium HJB equation is established as well.
