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Cet ouvrage propose une approche concise mais complete de la theorie de l'integrale stochastique dans le cadre general des semimartingales continues. Apres une introduction au mouvement brownien et a ses principales proprietes, les martingales et les semimartingales continues sont presentees en detail avant la construction de l'integrale stochastique. Les outils du calcul stochastique, incluant la formule d'Ito, le theoreme d'arret et de nombreuses applications, sont traites de maniere rigoureuse. Le livre contient aussi un chapitre sur les processus de Markov et un autre sur les equations differentielles stochastiques, avec une preuve detaillee des proprietes 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 Ito’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.
