By Gilles-Edouard Espinosa & Nizar Touzi (Cahier de la Chaire n°36)
Let X be a mean reverting scalar process, X the corresponding running maximum, T0 the first time X hits the level zero and ` a loss function, mainly increasing and convex. We consider the following optimal stopping problem (…) over all stopping times with values in [0; T0]. Under mild conditions, we prove that an optimal stopping time exists and is defined by (…) where the boundary is explicitly characterized as the concatenation of the solutions of two equations. We investigate some examples such as the Ornstein-Uhlenbeck process, the CIR-Feller process, as well as the standard and drifted Brownian motions. Finally, we perform an empirical examination of the efficiency of this strategy on real financial data.