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Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

Funded by The Danish National Research Foundation

MPS-RR 2000-48
December 2000

Solving the Poisson Disorder Problem


Goran Peskir

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 The key point in the solution is reached by specifying when the principle of smooth fit breaks down and gets superseded by the principle of continuous fit.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)