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**Optimal stopping rules.
Reprint of the 1978 3rd ed.**
*(English)*
Zbl 1138.60008

Applications of Mathematics 8. Berlin: Springer (ISBN 978-3-540-74010-0/pbk; 978-3-540-74011-7/ebook). xii, 217 p. (2008).

Optimal stopping problems form an important and well-developed class of stochastic control problems, where one needs to search for stopping times, at which the underlying stochastic processes should be stopped in order to optimize the values of given functionals. Such problems appear, for example, in stochastic calculus (maximal inequalities), mathematical statistics (sequential analysis) and finance (derivative pricing). Because of increasing demand of different applications in a great number of applied sciences, the optimal stopping theory remains now an intensively investigated research field. The methods of optimal stopping are now widely used in engineering, biological and life sciences, as well as in universe and earth sciences.

The book develops the general theory of optimal stopping problems for Markov processes in discrete and continuous time and illustrates its application on the problems of sequential analysis. Being the basic monograph in the field during the last three decades, the book has not lost its actuality. Further investigations on optimal stopping theory including the results of its connection with free-boundary problems and detailed analysis of principles of smooth and continuous fit can be found in the joined monograph of the author with G. Peskir [Optimal stopping and free-boundary problems (2006; Zbl 1115.60001)].

The book is stucturized as follows. Chapter 1 has a preparation character. It revises basic notions of probability theory, martingales and Markov processes in discrete and continuous time. Chapters 2 and 3 are the central part of the book. They deal with the construction of optimal stopping rules for Markov random sequences and processes. Although the results for discrete and continuous time cases are quite similar, their proofs are carried out in essentially different ways. Chapter 4 is devoted to the application of the theory to solving the problems of sequential statistics. The developed analytic tool is used to provide explicit solutions to the problems of sequential testing of two simple hypotheses and quickest disorder detection for an observed sequence of independent random variables and a linearly drifted Wiener process.

Two Russian editions of this book were published in 1969 (Zbl 0214.45301) and 1976 (Zbl 0463.62068). The corresponding English translations appeared in 1973 (Zbl 0267.62039) and 1978 (Zbl 0391.60002) (by AMS and Springer, respectively).

The book develops the general theory of optimal stopping problems for Markov processes in discrete and continuous time and illustrates its application on the problems of sequential analysis. Being the basic monograph in the field during the last three decades, the book has not lost its actuality. Further investigations on optimal stopping theory including the results of its connection with free-boundary problems and detailed analysis of principles of smooth and continuous fit can be found in the joined monograph of the author with G. Peskir [Optimal stopping and free-boundary problems (2006; Zbl 1115.60001)].

The book is stucturized as follows. Chapter 1 has a preparation character. It revises basic notions of probability theory, martingales and Markov processes in discrete and continuous time. Chapters 2 and 3 are the central part of the book. They deal with the construction of optimal stopping rules for Markov random sequences and processes. Although the results for discrete and continuous time cases are quite similar, their proofs are carried out in essentially different ways. Chapter 4 is devoted to the application of the theory to solving the problems of sequential statistics. The developed analytic tool is used to provide explicit solutions to the problems of sequential testing of two simple hypotheses and quickest disorder detection for an observed sequence of independent random variables and a linearly drifted Wiener process.

Two Russian editions of this book were published in 1969 (Zbl 0214.45301) and 1976 (Zbl 0463.62068). The corresponding English translations appeared in 1973 (Zbl 0267.62039) and 1978 (Zbl 0391.60002) (by AMS and Springer, respectively).

Reviewer: Pavel Gapeev (London)

### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60G40 | Stopping times; optimal stopping problems; gambling theory |

60J05 | Discrete-time Markov processes on general state spaces |

60J25 | Continuous-time Markov processes on general state spaces |

35R35 | Free boundary problems for PDEs |

62C10 | Bayesian problems; characterization of Bayes procedures |

62L15 | Optimal stopping in statistics |

62L10 | Sequential statistical analysis |