  We consider a non-Markovian optimal stopping problem on finite horizon. We
prove that the value process can be represented by means of a backward
stochastic differential equation (BSDE), defined on an enlarged probability
space, containing a stochastic integral having a one-jump point process as
integrator and an (unknown) process with a sign constraint as integrand. This
provides an alternative representation with respect to the classical one given
by 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 as
the value of an auxiliary optimization problem where the intensity of the point
process is controlled.
