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Stochastics
An International Journal of Probability and Stochastic Processes
Volume 79, 2007 - Issue 1-2: Optimal Stopping and Applications Part I
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Original Articles

Optimal stopping of the maximum process: a converse to the results of Peskir

Pages 85-102 | Received 02 May 2006, Accepted 29 Sep 2006, Published online: 05 Nov 2008
 

Abstract

Peskir, (and also Meilijson and Obłój) considered the following optimal stopping problem: find, for an increasing function F and a positive function λ,

where S is the maximum process of Brownian motion. In this article, we are interested in the converse: find, for an increasing function F and a suitable function λ,
In the non-degenerate cases the optimal stopping rule is of the form stop the first time that reaches γ or falls below where γ, a positive constant, and g, a negative function, are both to be chosen. The optimal function g is characterised as the solution to non-linear differential equation, which is very similar to that used by Peskir to characterise the solution to equation (Equation1), however we derive this differential equation in a completely different way.

2000 Mathematics Subject Classification::

Notes

The author is supported by an Epsrc Advanced Fellowship. The author thanks a pair of referees who provided detailed comments which helped improve the paper. Any remaining errors are the author's own responsibility.

Additional information

Notes on contributors

David Hobson

† †The author is supported by an Epsrc Advanced Fellowship. The author thanks a pair of referees who provided detailed comments which helped improve the paper. Any remaining errors are the author's own responsibility.

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