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**The Malliavin calculus and related topics.
2nd ed.**
*(English)*
Zbl 1099.60003

Probability and Its Applications. Berlin: Springer (ISBN 3-540-28328-5/hbk; 978-3-642-06651-1/pbk; 3-540-28329-3/ebook). xiv, 382 p. (2006).

The second edition appears ten years after the first one (1995; Zbl 0837.60050). For a detailed review of the first edition see [in: Stochastic processes and related topics, Math. Res. 61, 103–127 (1991; Zbl 0742.60055)]. The second edition includes two additional chapters on fractional Brownian motion and mathematical finance.

Publisher’s summary: The Malliavin calculus (or stochastic calculus of variations) is an infinite-dimensional differential calculus on a Gaussian space. Originally, it was developed to provide a probabilistic proof to Hörmander’s “sum of squares” theorem, but it has found a wide range of applications in stochastic analysis. This monograph presents the main features of the Malliavin calculus and discusses in detail its main applications. The author begins by developing the analysis on the Wiener space, and then uses this to establish the regularity of probability laws and to prove Hörmander’s theorem. The regularity of the law of stochastic partial differential equations driven by a space-time white noise is also studied. The subsequent chapters develop the connection of the Malliavin with the anticipating stochastic calculus, studying anticipating stochastic differential equations and the Markov property of solutions to stochastic differential equations with boundary conditions.

Publisher’s summary: The Malliavin calculus (or stochastic calculus of variations) is an infinite-dimensional differential calculus on a Gaussian space. Originally, it was developed to provide a probabilistic proof to Hörmander’s “sum of squares” theorem, but it has found a wide range of applications in stochastic analysis. This monograph presents the main features of the Malliavin calculus and discusses in detail its main applications. The author begins by developing the analysis on the Wiener space, and then uses this to establish the regularity of probability laws and to prove Hörmander’s theorem. The regularity of the law of stochastic partial differential equations driven by a space-time white noise is also studied. The subsequent chapters develop the connection of the Malliavin with the anticipating stochastic calculus, studying anticipating stochastic differential equations and the Markov property of solutions to stochastic differential equations with boundary conditions.

Reviewer: Stefan Weber (Ithaca)

### MSC:

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

60H05 | Stochastic integrals |

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

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

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

60H07 | Stochastic calculus of variations and the Malliavin calculus |