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## Über dieses Buch

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, which often 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.

## Inhaltsverzeichnis

### Chapter 1. Probabilistic-Statistical Models in Quickest Detection Problems. Discrete and Continuous Time

1. From the considerations presented in the introduction it follows that many probabilistic-statistical schemes like, for example, those used in production quality control, are such that in the behavior of the observable random sequences a “failure” occurs in the character of the probability distributions. In this section we will describe a sufficiently general scheme, called the “stochastic θ-model”, which also covers the cases considered above, in which θ is understood as the time at which the probability characteristics change (i.e., “disorder”, “disruption”, or “chaos” occurs).
Albert N. Shiryaev

### Chapter 2. Basic Settings and Solutions of Quickest Detection Problems. Discrete Time

1. At the outset we will assume that with the original probabilistic-statistical experiment $$(\Omega ,\mathcal {F}, (\mathcal {F}_t)_{t \geqslant 0};\mathrm {P}^{\kern 1pt 0}, \mathrm {P}^{\infty })$$, where
$$\displaystyle \{\varnothing ,\Omega \} =\mathcal {F}_0 \subseteq \mathcal {F}_1 \subseteq \cdots ,$$
there are associated “θ-models” and “G-models” (Sects. 1.​2 and 1.​3).
Albert N. Shiryaev

### Chapter 3. Optimal Stopping Times. General Theory for the Discrete-Time Case

1. A specific peculiarity of the quickest detection problems considered here is that in them one is required to determine a stopping time that is close, in some sense, to the “regime-failure” time in the observed process. For quite some time individual problems have emerged that effectively reduce to optimal stopping time problems and eventually lead to the construction of a theory of optimal stopping rules, the results of which are essential in dealing with the quickest detection problems we are considering.
Albert N. Shiryaev

### Chapter 4. Optimal Stopping Rules. General Theory for the Discrete-Time Case in the Markov Representation

1. Recall that the problems concerning optimal stopping and optimal stopping rules considered above were stated as follows.
Albert N. Shiryaev

### Chapter 5. Optimal Stopping Rules. General Theory for the Continuous-Time Case

1. We assume that there are given a filtered probability space $$(\Omega , \mathcal {F}, (\mathcal {F}_t)_{t \geqslant 0}, \mathrm {P})$$ and a family of stochastic processes $$H =(H_t)_{t \geqslant 0}$$, where H t will be interpreted as the payoff if the stopping occurs at the time t.
Albert N. Shiryaev

### Chapter 6. Basic Formulations and Solutions of Quickest Detection Problems. Continuous Time. Models with Brownian Motion

1. In the second chapter the A, B, C, and D variants of quickest detection problems were considered in the case of discrete time and arbitrary random sequences (on filtered probability spaces).
Albert N. Shiryaev

### Chapter 7. Multi-stage Quickest Detection of Breakdown of a Stationary Regime. Model with Brownian Motion

1. We will denote by θ the time of “disorder” onset (disorder time for short), assuming that $$0 \leqslant \theta \leqslant \infty$$, and by $$X=(X_t)_{t \geqslant 0}$$ the observed process with the stochastic differential
$$\displaystyle {} dX_t=\mu I (t \geqslant \theta ) \,dt+\sigma \,dB_t,\quad X_0=0.$$
Albert N. Shiryaev

### Chapter 8. Disorder on Filtered Probability Spaces

We assume again that at the basis of our considerations lies a filtered probabilistic-statistical experiment.
Albert N. Shiryaev

### Chapter 9. Bayesian and Variational Problems of Hypothesis Testing. Brownian Motion Models

1. Suppose that we observe the stochastic process $$X=(X_t)_{t \geqslant 0}$$,
$$\displaystyle X_t=\theta \mu t+B_t, \quad X_0=0,$$
on the filtered probability space , where $$B=(B_t)_{t \geqslant 0}$$ is the standard Brownian motion (as a martingale with respect to the filtration flow $$(\mathcal {F}_t)_{t \geqslant 0}$$, EB t = 0 and $$\mathrm {E} B_t^2=t$$).
Albert N. Shiryaev

### Chapter 10. Some Applications to Financial Mathematics

The parameter θ is the unknown disorder time, at which the drift coefficient of the process X changes its positive value μ 1 to a negative value μ 2. Accordingly, it is natural to call X a Brownian motion with change-of-trend disorder. (Related, but different from the one treated here, is the model where the volatility (σ) undergoes a jump. Such models are known as bubble models.) Another appropriate name for the model (10.1) is that of Bachelier model with disorder (with initial condition X 0 = 0; compare with [94, 95]).
Albert N. Shiryaev

### Backmatter

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