Author | Revuz, Daniel. author |
---|---|

Title | Continuous Martingales and Brownian Motion [electronic resource] / by Daniel Revuz, Marc Yor |

Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1991 |

Connect to | http://dx.doi.org/10.1007/978-3-662-21726-9 |

Descript | IX, 536 p. online resource |

SUMMARY

This book focuses on the probabilistic theory ofBrownian motion. This is a good topic to center a discussion around because Brownian motion is in the intersecยญ tioll of many fundamental classes of processes. It is a continuous martingale, a Gaussian process, a Markov process or more specifically a process with inยญ dependent increments; it can actually be defined, up to simple transformations, as the real-valued, centered process with independent increments and continuous paths. It is therefore no surprise that a vast array of techniques may be successยญ fully applied to its study and we, consequently, chose to organize the book in the following way. After a first chapter where Brownian motion is introduced, each of the following ones is devoted to a new technique or notion and to some of its applications to Brownian motion. Among these techniques, two are of paraยญ mount importance: stochastic calculus, the use ofwhich pervades the whole book and the powerful excursion theory, both of which are introduced in a selfยญ contained fashion and with a minimum of apparatus. They have made much easier the proofs of many results found in the epoch-making book of Itรถ and McKean: Diffusion Processes and their Sampie Paths, Springer (1965)

CONTENT

0. Preliminaries -- I. Introduction -- II. Martingales -- III. Markov Processes -- IV. Stochastic Integration -- V. Representation of Martingales -- VI. Local Times -- VII. Generators and Time Reversal -- VIII. Girsanovโ{128}{153}s Theorem and First Applications -- IX. Stochastic Differential Equations -- X. Additive Functionals of Brownian Motion -- XI. Bessel Processes and Ray-Knight Theorems -- XII. Excursions -- XIII. Limit Theorems in Distribution -- ยง 1. Gronwallโ{128}{153}s Lemma -- ยง 2. Distributions -- ยง 3. Convex Functions -- ยง 4. Hausdorff Measures and Dimension -- ยง 5. Ergodic Theory -- Index of Notation -- Index of Terms

Mathematics
Probabilities
Physics
Mathematics
Probability Theory and Stochastic Processes
Theoretical Mathematical and Computational Physics