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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.

Mathematical Statistics --- Mathematics --- Physical Sciences & Mathematics --- Distribution (Probability theory. --- Finance. --- Mathematics. --- Systems theory. --- Probability Theory and Stochastic Processes. --- Quantitative Finance. --- Measure and Integration. --- Mathematical Modeling and Industrial Mathematics. --- Systems Theory, Control. --- Funding --- Funds --- Economics --- Currency question --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Math --- Science --- Brownian motion processes. --- Martingales (Mathematics) --- Stochastic analysis. --- Probabilities. --- Economics, Mathematical . --- Measure theory. --- Mathematical models. --- System theory. --- Systems, Theory of --- Systems science --- Models, Mathematical --- Simulation methods --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Mathematical economics --- Econometrics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Philosophy --- Methodology --- Social sciences --- Control theory. --- Probability Theory. --- Mathematics in Business, Economics and Finance. --- Systems Theory, Control . --- Dynamics --- Machine theory

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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.

Mathematics --- Operational research. Game theory --- Probability theory --- waarschijnlijkheidstheorie --- stochastische analyse --- wiskunde --- kansrekening

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Branching processes --- Stochastic partial differential equations

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