Electronic Journal of Probability

Discounted optimal stopping for maxima in diffusion models with finite horizon

Pavel Gapeev

Full-text: Open access

Abstract

We present a solution to some discounted optimal stopping problem for the maximum of a geometric Brownian motion on a finite time interval. The method of proof is based on reducing the initial optimal stopping problem with the continuation region determined by an increasing continuous boundary surface to a parabolic free-boundary problem. Using the change-of-variable formula with local time on surfaces we show that the optimal boundary can be characterized as a unique solution of a nonlinear integral equation. The result can be interpreted as pricing American fixed-strike lookback option in a diffusion model with finite time horizon.

Article information

Source
Electron. J. Probab., Volume 11 (2006), paper no. 38, 1031-1048.

Dates
Accepted: 21 November 2006
First available in Project Euclid: 31 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464730573

Digital Object Identifier
doi:10.1214/EJP.v11-367

Mathematical Reviews number (MathSciNet)
MR2268535

Zentralblatt MATH identifier
1127.60037

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 35R35: Free boundary problems 45G10: Other nonlinear integral equations 60J60: Diffusion processes [See also 58J65] 91B28

Keywords
Discounted optimal stopping problem finite horizon geometric Brownian motion maximum process parabolic free-boundary problem smooth fit normal reflection a nonlinear Volterra integral equation of the second kind boundary surface a change-of-varia

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Gapeev, Pavel. Discounted optimal stopping for maxima in diffusion models with finite horizon. Electron. J. Probab. 11 (2006), paper no. 38, 1031--1048. doi:10.1214/EJP.v11-367. https://projecteuclid.org/euclid.ejp/1464730573


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