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We consider a class of infinite-time horizon optimal stopping problems forspectrally negative Levy processes. Focusing on strategies of threshold type,we write explicit expressions for the corresponding expected payoff via thescale function, and further pursue optimal candidate threshold levels. Weobtain and show the equivalence of the continuous/smooth fit condition and thefirst-order condition for maximization over threshold levels. As examples ofits applications, we give a short proof of the McKean optimal stopping problem(perpetual American put option) and solve an extension to Egami and Yamazaki(2013).

On the continuous and smooth fit principle for optimal stopping problems in spectrally negative Levy models