## Abstract

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 _{0}E[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 language | English (US) |
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Pages (from-to) | 2543-2572 |

Number of pages | 30 |

Journal | SIAM Journal on Control and Optimization |

Volume | 50 |

Issue number | 5 |

DOIs | |

State | Published - 2012 |

## Keywords

- 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