Detecting the maximum of a scalar diffusion with negative drift

Let X be a scalar diffusion process with drift coefficient pointing towards the origin, i.e. X is mean-reverting. We denote by X ^* the corresponding running maximum, T0 the first time X hits the level zero. Given an increasing and convex loss function l, we consider the following optimal stopping problem: 0≤θ≤T ₀E[l(X ^* _T0-X _θ)] , over all stopping times θ with values in [0, T ₀]. For the quadratic loss function and under mild conditions, we prove that an optimal stopping time exists and is defined by: θ ^* = T ₀ inf{t ≥ 0; X ^* _t ≥ γ(X _t)}, where the boundary γ 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.