Electronic Journal of Probability

Discounted optimal stopping for maxima in diffusion models with finite horizon

Pavel Gapeev

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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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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