1.

Record Nr.

UNINA9910254069803321

Autore

Le Gall Jean-François

Titolo

Brownian Motion, Martingales, and Stochastic Calculus [[electronic resource] /] / by Jean-François Le Gall

Pubbl/distr/stampa

Cham : , : Springer International Publishing : , : Imprint : Springer, , 2016

ISBN

3-319-31089-5

Edizione

[1st ed. 2016.]

Descrizione fisica

1 online resource (XIII, 273 p. 5 illus., 1 illus. in color.)

Collana

Graduate Texts in Mathematics, , 0072-5285 ; ; 274

Disciplina

519.23

Soggetti

Probabilities

Economics, Mathematical 

Measure theory

Mathematical models

System theory

Probability Theory and Stochastic Processes

Quantitative Finance

Measure and Integration

Mathematical Modeling and Industrial Mathematics

Systems Theory, Control

Lingua di pubblicazione

Inglese

Formato

Materiale a stampa

Livello bibliografico

Monografia

Note generali

Includes Index.

Nota di contenuto

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.

Sommario/riassunto

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.