Detecting the maximum of a scalar diffusion with negative drift

Gilles Edouard Espinosa, Nizar Touzi

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)2543-2572
Number of pages30
JournalSIAM Journal on Control and Optimization
Issue number5
StatePublished - 2012


  • Free-boundary problem
  • Markov process
  • Maximum process
  • Optimal stopping
  • Ordinary differential equation
  • Smooth fit
  • Verification argument

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics


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