## Continuous Martingales and Brownian MotionFrom the reviews: "This is a magnificent book! Its purpose is to describe in considerable detail a variety of techniques used by probabilists in the investigation of problems concerning Brownian motion. The great strength of Revuz and Yor is the enormous variety of calculations carried out both in the main text and also (by implication) in the exercises. ... This is THE book for a capable graduate student starting out on research in probability: the effect of working through it is as if the authors are sitting beside one, enthusiastically explaining the theory, presenting further developments as exercises, and throwing out challenging remarks about areas awaiting further research..." Bull.L.M.S. 24, 4 (1992) Since the first edition in 1991, an impressive variety of advances has been made in relation to the material of this book, and these are reflected in the successive editions. |

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### Contents

1 | |

2 | |

15 | |

3 Canonical Processes and Gaussian Processes | 33 |

Notes and Comments | 48 |

3 Optional Stopping Theorem | 68 |

2 Feller Processes | 88 |

3 Strong Markov Property | 102 |

Notes and Comments | 362 |

2 Existence and Uniqueness in the Case of Lipschitz Coefficients | 375 |

3 The Case of Hölder Coefficients in Dimension One | 388 |

Additive Functionals of Brownian Motion | 401 |

3 Ergodic Theorems for Additive Functionals | 422 |

4 Asymptotic Results for the Planar Brownian Motion | 430 |

Notes and Comments | 436 |

2 RayKnight Theorems | 454 |

Notes and Comments | 117 |

2 Stochastic Integrals | 137 |

3 Itôs Formula and First Applications | 146 |

4 BurkholderDavisGundy Inequalities | 160 |

Notes and Comments | 176 |

2 Conformal Martingales and Planar Brownian Motion | 189 |

4 Integral Representations | 209 |

2 The Local Time of Brownian Motion | 239 |

4 First Order Calculus | 260 |

Notes and Comments | 277 |

2 Diffusions and Itó Processes | 294 |

4 Time Reversal and Applications 3 13 | 321 |

2 Application of Girsanovs Theorem to the Study of Wieners Space | 338 |

3 Bessel Bridges | 463 |

Notes and Comments | 469 |

2 The Excursion Process of Brownian Motion | 480 |

3 Excursions Straddling a Given Time | 488 |

Notes and Comments | 511 |

2 Asymptotic Behavior of Additive Functionals of Brownian Motion | 522 |

3 Asymptotic Properties of Planar Brownian Motion | 531 |

Notes and Comments | 541 |

4 Hausdorff Measures and Dimension | 547 |

Index of Notation | 595 |

Catalogue | 605 |

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### Common terms and phrases

Bessel processes Borel function Brownian Bridge Brownian motion cadlag called Chap compact compute condition consequently constant cont continuous local martingale continuous semimartingale converges Corollary countable defined Definition denote density derivative distribution equal equation equivalent Exercise exists Feller process filtration finite variation function f Girsanov hence Hint Hölder continuous increasing process independent inequality inf{t interval Itô's formula Lebesgue measure Lemma Lévy processes Lévy's locally bounded Markov process Markov property mart monotone class theorem Moreover notation o-algebra o-field oo a.s. particular paths positive Borel function predictable process Prob probability measure probability space Proposition prove random variables reader real number Remark resp respect result right-continuous Sect semi-group sequence solution standard BM standard linear BM stochastic integral stopping strong Markov property submartingale subset time-change uniformly integrable unique vanishing zero