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In the spirit of [Surya07'], we develop an average problem approach to provethe optimality of threshold type strategies for optimal stopping of L\'evymodels with a continuous additive functional (CAF) discounting. Underspectrally negative models, we specialize this in terms of conditions on thereward function and random discounting, where we present two examples of localtime and occupation time discounting. We then apply this approach to recursiveoptimal stopping problems, and present simpler and neater proofs for a numberof important results on qualitative properties of the optimal thresholds, whichare only known under a few special cases.

On the optimality of threshold type strategies in single and recursive optimal stopping under L\'evy models