An optimal stopping problem with a reward constraint

This article studies an optimal stopping problem with an endogenous constraint on the set of admissible stopping times. The constraint stipulates that continuation is permitted, at any given date t, only if the endogenous reward achieved exceeds a prespecified threshold. Characterizations of the value function and the optimal stopping time are presented. An application to the pricing of corporate claims, when the capital structure of the firm includes equity-trigger debt, is carried out.