Le Gall, Jean-Francois
Brownian motion, martingales, and stochastic calculus - Switzerland Springer 2016 - xiii, 273p. - Graduate Texts in Mathematics 274 .
Includes References
Gaussian variables and Gaussian processes
Brownian motion
Filtrations and martingales
Continuous semimartingales
Stochastic integration
General theory of Markov processes
Brownian motion and partial differential equations
Stochastic differential equations
Local times
The monotone class lemma
Discrete martingales
References
This 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
Translated from the French edition published: Berlin: Springer, 2013
9783319310886 (HB)
Stochastic analysis--calculus
Brownian motion
Martingales
519.216 / LE
Brownian motion, martingales, and stochastic calculus - Switzerland Springer 2016 - xiii, 273p. - Graduate Texts in Mathematics 274 .
Includes References
Gaussian variables and Gaussian processes
Brownian motion
Filtrations and martingales
Continuous semimartingales
Stochastic integration
General theory of Markov processes
Brownian motion and partial differential equations
Stochastic differential equations
Local times
The monotone class lemma
Discrete martingales
References
This 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
Translated from the French edition published: Berlin: Springer, 2013
9783319310886 (HB)
Stochastic analysis--calculus
Brownian motion
Martingales
519.216 / LE