  In the spirit of [Surya07'], we develop an average problem approach to prove
the optimality of threshold type strategies for optimal stopping of L\'evy
models with a continuous additive functional (CAF) discounting. Under
spectrally negative models, we specialize this in terms of conditions on the
reward function and random discounting, where we present two examples of local
time and occupation time discounting. We then apply this approach to recursive
optimal stopping problems, and present simpler and neater proofs for a number
of important results on qualitative properties of the optimal thresholds, which
are only known under a few special cases.
