MPS-RR 2000-48

December 2000

# Solving the Poisson Disorder Problem

by:

### Albert N. Shiryaev

The Poisson disorder problem seeks to determine a stopping time
which is as close as possible to the (unknown) time of 'disorder'
when the intensity of an observed Poisson process changes from
one value to another. Partial answers to this question are known
to date only in some special cases, and the main purpose of the
present paper is to describe the structure of the solution in the
general case. The method of proof consists of reducing the initial
(optimal stopping) problem to a free-boundary differential-difference
problem. The key point in the solution is reached by specifying
when the principle of smooth fit breaks down and gets superseded
by the principle of continuous fit. This can be done in probabilistic
terms (by describing the sample path behaviour of the a posteriori
probability process) and in analytic terms (via the existence of
a singularity point of the free-boundary equation).

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This paper has now been published in *Adv.in Finance and Stochastics: Essays in Honour of Dieter Sondermann (invited paper), Springer-Verlag, 2002, (295-312)*