  We present a pair of adjoint optimal control problems characterizing a class
of time-symmetric stochastic processes defined on random time intervals. The
associated PDEs are of free-boundary type. The particularity of our approach is
that it involves two adjoint optimal stopping times adapted to a pair of
filtrations, the traditional increasing one and another, decreasing. They are
the keys of the time symmetry of the construction, which can be regarded as a
generalization of "Schr\"odinger's problem" (1931-32) to space-time domains.
The relation with the notion of "Hidden diffusions" is also described.
