An optimal stopping problem for fragmentation processes

TL;DR

This paper explores an optimal stopping problem within fragmentation processes, utilizing stopping lines and generalized Ornstein-Uhlenbeck processes, and solves it through classical verification techniques and exponential Levy process theory.

Contribution

It introduces a novel approach to formulate and solve an optimal stopping problem in fragmentation processes using stopping lines and Levy process techniques.

Findings

Reduction of the fragmentation stopping problem to a classical optimal stopping problem.

Application of modern Levy process theory to solve the problem.

Explicit solution obtained via verification techniques.

Abstract

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.