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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 Lévy processes.
Kyprianou, A. E., & Pardo, J. C. (2012). An optimal stopping problem for fragmentation processes. Stochastic Processes and their Applications, 122(4), 1210-1225. https://doi.org/10.1016/j.spa.2011.12.009