An optimal stopping problem for fragmentation processes (Q424467)

This paper addresses the previously never treated issue of how a homogeneous fragmentation process (for these, see, e.g., the book by \textit{J. Bertoin} [Random fragmentation and coagulation processes. Cambridge: Cambridge University Press (2006; Zbl 1107.60002)]) may be used to drive optimal stopping problems. In Section 3, the authors give (and interpret) a toy optimal stopping problem for a certain homogeneous fragmentation process. In Section 2, the definition of a stopping line (which was introduced in the book by Bertoin [loc. cit.]) is given and the ``many-to-one principle'' is formulated which basically goes back to \textit{J. Berestycki}, \textit{S. C. Harris} and \textit{A. E. Kyprianou} [Ann. Appl. Probab. 21, No. 5, 1749--1794 (2011; Zbl 1245.60069)]. It is then shown how the theory of stopping lines allows the above optimal stopping problem to be converted to a classical optimal stopping problem for a generalized Ornstein-Uhlenbeck process associated with Bertoin's tagged fragment. The latter is then solved by using a classical verification technique thanks to the application of certain aspects of the modern theory of integrated exponential Lévy processes.