04544nam 22007935 450 991025406980332120230810185430.03-319-31089-510.1007/978-3-319-31089-3(CKB)3710000000653684(SSID)ssj0001665922(PQKBManifestationID)16455552(PQKBTitleCode)TC0001665922(PQKBWorkID)15000904(PQKB)11344602(DE-He213)978-3-319-31089-3(MiAaPQ)EBC6311722(MiAaPQ)EBC5586601(Au-PeEL)EBL5586601(OCoLC)1066198180(PPN)193444348(EXLCZ)99371000000065368420160428d2016 u| 0engurnn|008mamaatxtccrBrownian Motion, Martingales, and Stochastic Calculus /by Jean-François Le Gall1st ed. 2016.Cham :Springer International Publishing :Imprint: Springer,2016.1 online resource (XIII, 273 p. 5 illus., 1 illus. in color.) Graduate Texts in Mathematics,2197-5612 ;274Includes Index.3-319-31088-7 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.Graduate Texts in Mathematics,2197-5612 ;274ProbabilitiesSocial sciencesMathematicsMeasure theoryMathematical modelsSystem theoryControl theoryProbability TheoryMathematics in Business, Economics and FinanceMeasure and IntegrationMathematical Modeling and Industrial MathematicsSystems Theory, Control Probabilities.Social sciencesMathematics.Measure theory.Mathematical models.System theory.Control theory.Probability Theory.Mathematics in Business, Economics and Finance.Measure and Integration.Mathematical Modeling and Industrial Mathematics.Systems Theory, Control .519.23Le Gall Jean-Françoisauthttp://id.loc.gov/vocabulary/relators/aut348889MiAaPQMiAaPQMiAaPQBOOK9910254069803321Brownian motion, martingales, and stochastic calculus1523199UNINA