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 , we consider the following optimal stopping problem: inf0≤θ≤T0 E[ (X ∗ T0 −Xθ)], over all stopping times θ with values in [0, T0]. For the quadratic loss function and under mild conditions, we prove that an optimal stopping time exists and is defined by: θ∗ = T0 ∧ inf{t ≥ 0; X∗ t ≥ γ(Xt)}, 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.