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
![](//:0)
reaches γ or
![](//:0)
falls below
![](//:0)
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![](//:0)
), 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.