LEADER 04544nam 22007935 450 001 9910254069803321 005 20230810185430.0 010 $a3-319-31089-5 024 7 $a10.1007/978-3-319-31089-3 035 $a(CKB)3710000000653684 035 $a(SSID)ssj0001665922 035 $a(PQKBManifestationID)16455552 035 $a(PQKBTitleCode)TC0001665922 035 $a(PQKBWorkID)15000904 035 $a(PQKB)11344602 035 $a(DE-He213)978-3-319-31089-3 035 $a(MiAaPQ)EBC6311722 035 $a(MiAaPQ)EBC5586601 035 $a(Au-PeEL)EBL5586601 035 $a(OCoLC)1066198180 035 $a(PPN)193444348 035 $a(EXLCZ)993710000000653684 100 $a20160428d2016 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt 182 $cc 183 $acr 200 10$aBrownian Motion, Martingales, and Stochastic Calculus /$fby Jean-François Le Gall 205 $a1st ed. 2016. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2016. 215 $a1 online resource (XIII, 273 p. 5 illus., 1 illus. in color.) 225 1 $aGraduate Texts in Mathematics,$x2197-5612 ;$v274 300 $aIncludes Index. 311 $a3-319-31088-7 327 $aGaussian 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. 330 $aThis book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Itô?s formula, the optional stopping theorem and Girsanov?s theorem, are treated in detail alongside many illustrative examples. The book also contains an introduction to Markov processes, with applications to solutions of stochastic differential equations and to connections between Brownian motion and partial differential equations. The theory of local times of semimartingales is discussed in the last chapter. Since its invention by Itô, stochastic calculus has proven to be one of the most important techniques of modern probability theory, and has been used in the most recent theoretical advances as well as in applications to other fields such as mathematical finance. Brownian Motion, Martingales, and Stochastic Calculus provides a strong theoretical background to the reader interested in such developments. Beginning graduate or advanced undergraduate students will benefit from this detailed approach to an essential area of probability theory. The emphasis is on concise and efficient presentation, without any concession to mathematical rigor. The material has been taught by the author for several years in graduate courses at two of the most prestigious French universities. The fact that proofs are given with full details makes the book particularly suitable for self-study. The numerous exercises help the reader to get acquainted with the tools of stochastic calculus. 410 0$aGraduate Texts in Mathematics,$x2197-5612 ;$v274 606 $aProbabilities 606 $aSocial sciences$xMathematics 606 $aMeasure theory 606 $aMathematical models 606 $aSystem theory 606 $aControl theory 606 $aProbability Theory 606 $aMathematics in Business, Economics and Finance 606 $aMeasure and Integration 606 $aMathematical Modeling and Industrial Mathematics 606 $aSystems Theory, Control 615 0$aProbabilities. 615 0$aSocial sciences$xMathematics. 615 0$aMeasure theory. 615 0$aMathematical models. 615 0$aSystem theory. 615 0$aControl theory. 615 14$aProbability Theory. 615 24$aMathematics in Business, Economics and Finance. 615 24$aMeasure and Integration. 615 24$aMathematical Modeling and Industrial Mathematics. 615 24$aSystems Theory, Control . 676 $a519.23 700 $aLe Gall$b Jean-François$4aut$4http://id.loc.gov/vocabulary/relators/aut$0348889 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910254069803321 996 $aBrownian motion, martingales, and stochastic calculus$91523199 997 $aUNINA