Change of time and change of measure.

*(English)*Zbl 1234.60003
Advanced Series on Statistical Science & Applied Probability 13. Hackensack, NJ: World Scientific (ISBN 978-981-4324-47-2/hbk; 978-981-4343-54-1/ebook). xvi, 305 p. (2010).

This book provides a comprehensive account of two topics that are of particular significance in both theoretical and applied stochastics: random change of time and change of probability law. The random change of time turns out to be a crucial tool for understanding the nature of various stochastic processes, and gives rise to interesting mathematical results and insights of importance for the modelling and interpretation of empirically observed dynamic processes. The change of probability law has been proved to be an absolutely necessary technique for solving central questions in mathematical finance, and also has a considerable role in insurance mathematics, large deviation theory and other fields. The book comprehensively collects and integrates results from a number of scattered sources in the literature and discusses the importance of the results relative to the existing literature, particularly with regard to mathematical finance. It is invaluable as a textbook for graduate-level courses and students or a handy reference for researchers and practitioners in financial mathematics and econometrics.

The book consists of twelve chapters. Chapter 1 is devoted to basic properties and representations of random change of time. Chapter 2 contains stochastic integral representations and application of change of time in stochastic integrals. Chapter 3 contains basic notions and elements of stochastic analysis in a general semimartingale framework. Chapter 4 is devoted to the properties of stochastic exponential, stochastic logarithm, and cumulant process. Chapter 5 contains a review of processes with independent increments and Lévy processes. Chapter 6 is devoted to the general techniques of change of measure. Chapter 7 contains applications of change of measure in models based on Lévy processes. Chapter 8 contains applications of change of measure in the semimartingale models and models based on Brownian motion and Lévy processes. Chapter 9 is devoted to conditionally Gaussian distributions and stochastic volatility models for the discrete-time case. Chapter 10 is devoted to martingale measures in the stochastic theory of arbitrage. Chapter 11 is devoted to change of measure in option pricing. Chapter 12 is devoted to conditionally Brownian and Lévy processes and stochastic volatility models.

The book consists of twelve chapters. Chapter 1 is devoted to basic properties and representations of random change of time. Chapter 2 contains stochastic integral representations and application of change of time in stochastic integrals. Chapter 3 contains basic notions and elements of stochastic analysis in a general semimartingale framework. Chapter 4 is devoted to the properties of stochastic exponential, stochastic logarithm, and cumulant process. Chapter 5 contains a review of processes with independent increments and Lévy processes. Chapter 6 is devoted to the general techniques of change of measure. Chapter 7 contains applications of change of measure in models based on Lévy processes. Chapter 8 contains applications of change of measure in the semimartingale models and models based on Brownian motion and Lévy processes. Chapter 9 is devoted to conditionally Gaussian distributions and stochastic volatility models for the discrete-time case. Chapter 10 is devoted to martingale measures in the stochastic theory of arbitrage. Chapter 11 is devoted to change of measure in option pricing. Chapter 12 is devoted to conditionally Brownian and Lévy processes and stochastic volatility models.

Reviewer: Pavel Gapeev (London)

##### MSC:

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

60H05 | Stochastic integrals |

60G10 | Stationary stochastic processes |

60G18 | Self-similar stochastic processes |

60G44 | Martingales with continuous parameter |

60G48 | Generalizations of martingales |

60G52 | Stable stochastic processes |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

60J65 | Brownian motion |

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

91B70 | Stochastic models in economics |