LEADER 04054nam 2200529 450 001 9910554280303321 005 20230629233334.0 010 $a3-11-074127-X 024 7 $a10.1515/9783110741278 035 $a(CKB)5590000000554961 035 $a(MiAaPQ)EBC6739364 035 $a(Au-PeEL)EBL6739364 035 $a(OCoLC)1266228052 035 $a(DE-B1597)576983 035 $a(DE-B1597)9783110741278 035 $a(EXLCZ)995590000000554961 100 $a20220625d2021 uy 0 101 0 $aeng 135 $aurcn#---||||| 181 $ctxt$2rdacontent 182 $cn$2rdamedia 183 $acr$2rdacarrier 200 10$aBrownian Motion $ea guide to random processes and stochastic calculus with a chapter on simulation by bjo?rn bo?ttcher /$fRene? L. Schilling 205 $aSecond edition. 210 1$aBoston, Massachusetts :$cDe Gruyter,$d[2021] 210 4$d©2021 215 $a1 online resource $cillustrations 225 1 $aDe Gruyter textbook 300 $aIncludes bibliographical references and index. 311 1 $a3-11-074125-3 327 $aIntro -- Preface -- Contents -- Dependence chart -- 1 Robert Brown's new thing -- 2 Brownian motion as a Gaussian process -- 3 Constructions of Brownian motion -- 4 The canonical model -- 5 Brownian motion as a martingale -- 6 Brownian motion as a Markov process -- 7 Brownian motion and transition semigroups -- 8 The PDE connection -- 9 The variation of Brownian paths -- 10 Regularity of Brownian paths -- 11 Brownian motion as a random fractal -- 12 The growth of Brownian paths -- 13 Strassen's functional law of the iterated logarithm -- 14 Skorokhod representation -- 15 Stochastic integrals: L< -- sup> -- 2< -- /sup> -- -Theory -- 16 Stochastic integrals: localization -- 17 Stochastic integrals: martingale drivers -- 18 Itô's formula -- 19 Applications of Itô's formula -- 20 Wiener Chaos and iterated Wiener-Itô integrals -- 21 Stochastic differential equations -- 22 Stratonovich's stochastic calculus -- 23 On diffusions -- 24 Simulation of Brownian motion by Björn Böttcher -- A Appendix -- Bibliography -- Index. 330 $aBrownian motion is one of the most important stochastic processes in continuous time and with continuous state space. Within the realm of stochastic processes, Brownian motion is at the intersection of Gaussian processes, martingales, Markov processes, diffusions and random fractals, and it has influenced the study of these topics. Its central position within mathematics is matched by numerous applications in science, engineering and mathematical finance. Often textbooks on probability theory cover, if at all, Brownian motion only briefly. On the other hand, there is a considerable gap to more specialized texts on Brownian motion which is not so easy to overcome for the novice. The authors' aim was to write a book which can be used as an introduction to Brownian motion and stochastic calculus, and as a first course in continuous-time and continuous-state Markov processes. They also wanted to have a text which would be both a readily accessible mathematical back-up for contemporary applications (such as mathematical finance) and a foundation to get easy access to advanced monographs. This textbook, tailored to the needs of graduate and advanced undergraduate students, covers Brownian motion, starting from its elementary properties, certain distributional aspects, path properties, and leading to stochastic calculus based on Brownian motion. It also includes numerical recipes for the simulation of Brownian motion. 410 0$aDe Gruyter textbook 606 $aBrownian motion processes 606 $aStochastic processes 615 0$aBrownian motion processes. 615 0$aStochastic processes. 676 $a519.233 686 $aSK 820$qSEPA$2rvk 700 $aSchilling$b Rene? L.$0478394 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910554280303321 996 $aBrownian Motion$92815963 997 $aUNINA