## Brownian Motion, Martingales, and Stochastic CalculusThis book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Itô’s formula, the optional stopping theorem and Girsanov’s theorem, are treated in detail alongside many illustrative examples. The book also contains an introduction to Markov processes, with applications to solutions of stochastic differential equations and to connections between Brownian motion and partial differential equations. The theory of local times of semimartingales is discussed in the last chapter. Since its invention by Itô, stochastic calculus has proven to be one of the most important techniques of modern probability theory, and has been used in the most recent theoretical advances as well as in applications to other fields such as mathematical finance. Brownian Motion, Martingales, and Stochastic Calculus provides a strong theoretical background to the reader interested in such developments.Beginning graduate or advanced undergraduate students will benefit from this detailed approach to an essential area of probability theory. The emphasis is on concise and efficient presentation, without any concession to mathematical rigor. The material has been taught by the author for several years in graduate courses at two of the most prestigious French universities. The fact that proofs are given with full details makes the book particularly suitable for self-study. The numerous exercises help the reader to get acquainted with the tools of stochastic calculus. |

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### Contents

1 | |

2 Brownian Motion | 18 |

3 Filtrations and Martingales | 41 |

4 Continuous Semimartingales | 69 |

5 Stochastic Integration | 97 |

6 General Theory of Markov Processes | 151 |

7 Brownian Motion and Partial Differential Equations | 185 |

8 Stochastic Differential Equations | 209 |

### Other editions - View all

Brownian Motion, Martingales, and Stochastic Calculus Jean-Francois Le Gall No preview available - 2018 |

Brownian Motion, Martingales, and Stochastic Calculus Jean-François Le Gall No preview available - 2016 |

### Common terms and phrases

Ä tg apply assume bounded Brownian motion started C.RC càdlàg canonical filtration centered Gaussian chapter Co(E completes the proof continuous function continuous local martingale continuous sample paths continuous semimartingale Corollary d-dimensional Brownian motion defined definition desired result Dirichlet problem distribution Exercise exists filtered probability space finite variation process fixed follows Ft/-Brownian motion Gaussian process hence hM;Mi hM;Ni holds implies independent inequality inf{t Itô’s formula lemma linear mapping Markov process Markov property measurable function monotone class nonnegative notation O-field optional stopping theorem planar Brownian motion pre-Brownian motion probability measure Proposition random variable real Brownian motion right-continuous semigroup sequence Show ſº stochastic calculus stochastic differential equations stochastic integral strong Markov property subset supermartingale twice continuously differentiable uniformly integrable uniformly integrable martingale uniqueness values vector verify