  The scope of this paper is to study the optimal stopping problems associated
to a stochastic process, which may represent the gain of an investment, for
which information on the final value is available a priori. This information
may proceed, for example, from insider trading or from pinning at expiration of
stock options. We solve and provide explicit solutions to these optimization
problems. As special case, we discuss different processes whose optimal barrier
has the same shape as the optimal barrier of the Brownian bridge. So doing we
provide a catalogue of alternatives to the Brownian bridge which in practice
could be better adapted to the data. Moreover, we investigate if, for any given
(decreasing) curve, there exists a process with this curve as optimal barrier.
This provides a model for the optimal liquidation time, i.e. the optimal time
at which the investor should liquidate a position in order to maximize the
gain.
