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Optimal Stopping of the Maximum Process

Part of: Stochastic processes Markov processes Sequential methods

Published online by Cambridge University Press:  30 January 2018

Luis H. R. Alvarez  and
Pekka Matomäki
Luis H. R. Alvarez*
Affiliation:
University of Turku
Pekka Matomäki*
Affiliation:
University of Turku
*
∗ Postal address: Department of Accounting and Finance, Turku School of Economics, University of Turku, Turku, FI-20500, Finland.
∗∗ Email address: lhralv@utu.fi
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Abstract

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We consider a class of optimal stopping problems involving both the running maximum as well as the prevailing state of a linear diffusion. Instead of tackling the problem directly via the standard free boundary approach, we take an alternative route and present a parameterized family of standard stopping problems of the underlying diffusion. We apply this family to delineate circumstances under which the original problem admits a unique, well-defined solution. We then develop a discretized approach resulting in a numerical algorithm for solving the considered class of stopping problems. We illustrate the use of the algorithm in both a geometric Brownian motion and a mean reverting diffusion setting.

Keywords

Optimal stoppinglinear diffusionsmaximum process

MSC classification

Primary: 60J60: Diffusion processes 62L15: Optimal stopping
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 818 - 836
Copyright
© Applied Probability Trust 

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