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**Optimal stopping and free-boundary problems.**
*(English)*
Zbl 1115.60001

Lectures in Mathematics, ETH Zürich. Basel: Birkhäuser (ISBN 3-7643-2419-8/hbk). xxii, 500 p. (2006).

The book discloses a fascinating connection between optimal stopping problems in probability and free-boundary problems in analysis, and focuses on key examples. The general theory of optimal stopping is exposed at the level of its basic principles in both discrete and continuous time covering martingale and Markovian methods. Methods of solution explained range from change of time, change of space, and change of measure, to more recent ones such as local time-space calculus and nonlinear integral equations. A detailed chapter on stochastic processes makes the material more accessible to a wider cross-disciplinary audience. The book will appeal to those wishing to master stochastic calculus via fundamental examples. Areas of application include financial mathematics, financial engineering, mathematical statistics, and stochastic analysis.

The book consists of eight chapters. Chapter 1 is devoted to general facts of optimal stopping in discrete and continuous time comparing martingale and Markovian approaches. Chapter 2 contains a brief review of stochastic processes including martingales and Markov processes, as well as of basic transformations: change of time, change of space, and change of measure. Chapter 3 is devoted to the connection of optimal stopping problems and free-boundary problems and connections with the Kolmogorov backward equations. Chapter 4 contains the methods of solutions of optimal stopping problems: reduction to free-boundary problems, superharmonic characterization of the value function, the method of time change, the method of space change, and the method of measure change; as well as optimal stopping problems for the maximum process and nonlinear integral equations. Chapter 5 is devoted to optimal stopping problems in stochastic analysis: Wald inequalities, Bessel inequalities, Doob inequalities, Hardy-Littlewood inequalities, and a brief summary of Burkholder-Davis-Gundy inequalities. Chapter 6 is devoted to optimal stopping problems in mathematical statistics: sequential testing problem for Wiener process with infinite and finite horizon and for Poisson process with infinite horizon. Chapter 7 is devoted to optimal stopping problems in mathematical finance: the American option problem, the Russian option problem, and the Asian option problem in Black-Merton-Scholes problem with infinite and finite time horizon. Chapter 8 is devoted to optimal stopping problems in financial engineering, namely, to the problem of stopping Brownian motion (with constant drift) without anticipation as close as possible to its ultimate position, ultimate integral, and ultimate maximum.

The book consists of eight chapters. Chapter 1 is devoted to general facts of optimal stopping in discrete and continuous time comparing martingale and Markovian approaches. Chapter 2 contains a brief review of stochastic processes including martingales and Markov processes, as well as of basic transformations: change of time, change of space, and change of measure. Chapter 3 is devoted to the connection of optimal stopping problems and free-boundary problems and connections with the Kolmogorov backward equations. Chapter 4 contains the methods of solutions of optimal stopping problems: reduction to free-boundary problems, superharmonic characterization of the value function, the method of time change, the method of space change, and the method of measure change; as well as optimal stopping problems for the maximum process and nonlinear integral equations. Chapter 5 is devoted to optimal stopping problems in stochastic analysis: Wald inequalities, Bessel inequalities, Doob inequalities, Hardy-Littlewood inequalities, and a brief summary of Burkholder-Davis-Gundy inequalities. Chapter 6 is devoted to optimal stopping problems in mathematical statistics: sequential testing problem for Wiener process with infinite and finite horizon and for Poisson process with infinite horizon. Chapter 7 is devoted to optimal stopping problems in mathematical finance: the American option problem, the Russian option problem, and the Asian option problem in Black-Merton-Scholes problem with infinite and finite time horizon. Chapter 8 is devoted to optimal stopping problems in financial engineering, namely, to the problem of stopping Brownian motion (with constant drift) without anticipation as close as possible to its ultimate position, ultimate integral, and ultimate maximum.

Reviewer: Pavel Gapeev (Berlin)

### MSC:

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

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

35R35 | Free boundary problems for PDEs |

45G10 | Other nonlinear integral equations |

45G15 | Systems of nonlinear integral equations |

62C10 | Bayesian problems; characterization of Bayes procedures |

62L15 | Optimal stopping in statistics |

60H05 | Stochastic integrals |

91B28 | Finance etc. (MSC2000) |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |