Stochastic Disorder Problems

Description

This monograph focuses on those stochastic quickest detection tasks in disorder problems that arise in the dynamical analysis of statistical data. These include quickest detection of randomly appearing targets, of spontaneously arising effects, and of arbitrage (in financial mathematics). There is also currently great interest in quickest detection methods for randomly occurring intrusions in information systems and in the design of defense methods against cyber-attacks. The author shows that the majority of quickest detection problems can be reformulated as optimal stopping problems where the stopping time is the moment the occurrence of disorder is signaled. Thus, considerable attention is devoted to the general theory of optimal stopping rules, and to its concrete problem-solving methods. The exposition covers both the discrete time case, which is in principle relatively simple and allows step-by-step considerations, and the continuous-time case,whichoften requires more technical machinery such as martingales, supermartingales, and stochastic integrals. There is a focus on the well-developed apparatus of Brownian motion, which enables the exact solution of many problems. The last chapter presents applications to financial markets. Researchers and graduate students interested in probability, decision theory and statistical sequential analysis will find this book useful. Front Matter ....Pages i-xix Probabilistic-Statistical Models in Quickest Detection Problems. Discrete and Continuous Time (Albert N. Shiryaev)....Pages 1-12 Basic Settings and Solutions of Quickest Detection Problems. Discrete Time (Albert N. Shiryaev)....Pages 13-56 Optimal Stopping Times. General Theory for the Discrete-Time Case (Albert N. Shiryaev)....Pages 57-73 Optimal Stopping Rules. General Theory for the Discrete-Time Case in the Markov Representation (Albert N. Shiryaev)....Pages 75-91 Optimal Stopping Rules. General Theory for the Continuous-Time Case (Albert N. Shiryaev)....Pages 93-137 Basic Formulations and Solutions of Quickest Detection Problems. Continuous Time. Models with Brownian Motion (Albert N. Shiryaev)....Pages 139-216 Multi-stage Quickest Detection of Breakdown of a Stationary Regime. Model with Brownian Motion (Albert N. Shiryaev)....Pages 217-237 Disorder on Filtered Probability Spaces (Albert N. Shiryaev)....Pages 239-275 Bayesian and Variational Problems of Hypothesis Testing. Brownian Motion Models (Albert N. Shiryaev)....Pages 277-366 Some Applications to Financial Mathematics (Albert N. Shiryaev)....Pages 367-388 Back Matter ....Pages 389-397