  We consider Backward Stochastic Differential Equations in a setting where
noise is generated by a countable state, continuous time Markov chain, and the
terminal value is prescribed at a stopping time. We show that, given sufficient
integrability of the stopping time and a growth bound on the terminal value and
BSDE driver, these equations admit unique solutions satisfying the same growth
bound (up to multiplication by a constant). This holds without assuming that
the driver is monotone in y, that is, our results do not require that the
terminal value be discounted at some uniform rate. We show that the conditions
are satisfied for hitting times of states of the chain, and hence present some
novel applications of the theory of these BSDEs.
