Stochastic Disorder Problems

I found the title of this book a little confusing. When I think of disorder I think of systems that have a random environment, but using google and consulting Wikipedia, I find that I am wrong. Stochastic disorder problems involve the detection of abrupt changes in the statistical behavior of a sequence of observations. They are also known as quickest detection problems since the goal is to minimize the time from change to detection. A classical example first considered in the 1920's and1930's is the quality control of a manufacturing process. However, there is also interest in quickest detection methods for randomly occurring intrusions into information systems and in the design of defense methods against cyber-attacks. 

 

Chapters 1 and 2 define the problem in discrete and continuous time. The discrete case is relatively simple thanks to the availability of recurrence relations and backward induction. The continuous case is more complicated involving the use of martingales, supermartingales, Brownian motion, stochastic integrals, etc.  Chapters 3-5 develop the main tool, optimal stopping rules.  The final five chapters consider different classes of problems, culminating with some applications to finance. Throughout this systematic treatment, four variants of the problem are considered. Two are Bayesian and two are minimax.

 

Shiryaev has written about two dozen books covering many topics in probability and statistics. In this book and all of his others, the style is precise and scholarly and the reader will appreciate the care with which he leads us through the often deep ideas behind the subject. Researchers and graduate students interested in optimal stopping, decision theory, and statistical sequential analysis will find this book useful. It is a very welcome addition to the literature of these fields.

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Rick Durrett is a professor in the Mathematics Department at Duke University.