  In this article we consider a toy example of an optimal stopping problem
driven by fragmentation processes. We show that one can work with the concept
of stopping lines to formulate the notion of an optimal stopping problem and
moreover, to reduce it to a classical optimal stopping problem for a
generalized Ornstein-Uhlenbeck process associated with Bertoin's tagged
fragment. We go on to solve the latter using a classical verification technique
thanks to the application of aspects of the modern theory of integrated
exponential Levy processes.
