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We consider a non-Markovian optimal stopping problem on finite horizon. Weprove that the value process can be represented by means of a backwardstochastic differential equation (BSDE), defined on an enlarged probabilityspace, containing a stochastic integral having a one-jump point process asintegrator and an (unknown) process with a sign constraint as integrand. Thisprovides an alternative representation with respect to the classical one givenby a reflected BSDE. The connection between the two BSDEs is also clarified.Finally, we prove that the value of the optimal stopping problem is the same asthe value of an auxiliary optimization problem where the intensity of the pointprocess is controlled.

Non-Markovian optimal stopping problems and constrained BSDEs with jump