Compound Poisson Disorder Problems with Nonlinear Detection Delay Penalty Cost Functions

Abstract

The quickest detection of the unknown and unobservable disorder time, when the arrival rate and mark distribution of a compound Poisson process suddenly changes, is formulated in a Bayesian setting, where the detection delay penalty is a general smooth function of the detection delay time. Under suitable conditions, the problem is shown to be equivalent to the optimal stopping of a finite-dimensional piecewise-deterministic strongly Markov sufficient statistic. The solution of the optimal stopping problem is described in detail for the compound Poisson disorder problem with polynomial detection delay penalty function of arbitrary but fixed degree. The results are illustrated for the case of the quadratic detection delay penalty function.

Keywords:

• Bayesian sequential change detection
• Compound Poisson disorder problem
• Optimal stopping
• Piecewise-deterministic Markov processes

Subject Classifications:

• 62L10
• 62L15
• 62C10
• 60G40

ACKNOWLEDGMENTS

The author thanks the anonymous referee and the editor Alexander Tartakovsky for thoughtful remarks and valuable suggestions, which improved the exposition of the paper.

Notes

Recommended by A. G. Tartakovsky