Jean Franc ois Le Gall Livres

Cet ouvrage propose une approche concise mais complète de la théorie de l'intégrale stochastique dans le cadre général des semimartingales continues. Après une introduction au mouvement brownien et à ses principales propriétés, les martingales et les semimartingales continues sont présentées en détail avant la construction de l'intégrale stochastique. Les outils du calcul stochastique, incluant la formule d'Itô, le théorème d'arrêt et de nombreuses applications, sont traités de manière rigoureuse. Le livre contient aussi un chapitre sur les processus de Markov et un autre sur les équations différentielles stochastiques, avec une preuve détaillée des propriétés markoviennes des solutions. De nombreux exercices permettent au lecteur de se familiariser avec les techniques du calcul stochastique.This book offers a rigorous and self-contained approach to the theory 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 the Girsanov theorem are treated in detail including many important applications. Two chapters are devoted to general Markov processes and to stochastic differential equations, with a complete derivation of Markovian properties of solutions in the Lipschitz case. Numerous exercises help the reader to get acquainted with the techniques of stochastic calculus.

This book presents a rigorous and self-contained exploration of stochastic integration and calculus within the framework of continuous semimartingales. It thoroughly covers essential tools of stochastic calculus, such as Itô’s formula, the optional stopping theorem, and Girsanov’s theorem, accompanied by numerous illustrative examples. An introduction to Markov processes is included, highlighting applications to stochastic differential equations and the relationship between Brownian motion and partial differential equations. The final chapter discusses the theory of local times of semimartingales. Since its inception by Itô, stochastic calculus has become a crucial technique in modern probability theory, influencing both theoretical advancements and practical applications, particularly in mathematical finance. This work provides a strong theoretical foundation for readers interested in these developments. It is tailored for beginning graduate or advanced undergraduate students, emphasizing concise and efficient presentation without sacrificing mathematical rigor. The author has taught this material in graduate courses at prestigious French universities, and the detailed proofs make it suitable for self-study. Additionally, numerous exercises help readers familiarize themselves with the tools of stochastic calculus.