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

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 (1), however we derive this differential equation in a completely different way.

Keywords:

• Optimal stopping
• Brownian motion
• Excursion theory
• Skorokhod embedding problem

2000 Mathematics Subject Classification::

• 62L17
• 60G40
• 60J65

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.