LEADER 08004nam 22007455 450 001 9910254067503321 005 20200629211338.0 010 $a3-319-27128-8 024 7 $a10.1007/978-3-319-27128-6 035 $a(CKB)3710000000627466 035 $a(SSID)ssj0001660995 035 $a(PQKBManifestationID)16437999 035 $a(PQKBTitleCode)TC0001660995 035 $a(PQKBWorkID)14989309 035 $a(PQKB)11322511 035 $a(DE-He213)978-3-319-27128-6 035 $a(MiAaPQ)EBC6314232 035 $a(MiAaPQ)EBC5591397 035 $a(Au-PeEL)EBL5591397 035 $a(OCoLC)1066194701 035 $a(PPN)19277221X 035 $a(EXLCZ)993710000000627466 100 $a20160311d2016 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt 182 $cc 183 $acr 200 10$aStochastic Integration by Parts and Functional Itô Calculus /$fby Vlad Bally, Lucia Caramellino, Rama Cont ; edited by Frederic Utzet, Josep Vives 205 $a1st ed. 2016. 210 1$aCham :$cSpringer International Publishing :$cImprint: Birkhäuser,$d2016. 215 $a1 online resource (IX, 207 p. 1 illus. in color.) 225 1 $aAdvanced Courses in Mathematics - CRM Barcelona,$x2297-0304 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a3-319-27127-X 327 $aIntro -- Foreword -- Contents -- Part I Integration by Parts Formulas, Malliavin Calculus, and Regularity of Probability Laws -- Preface -- Problem 1 -- Problem 2 -- Problem 3 -- Problem 4 -- Conclusion -- Chapter 1 Integration by parts formulas and the Riesz transform -- 1.1 Sobolev spaces associated to probability measures -- 1.2 The Riesz transform -- 1.3 A first absolute continuity criterion: Malliavin-Thalmaier representation formula -- 1.4 Estimate of the Riesz transform -- 1.5 Regularity of the density -- 1.6 Estimate of the tails of the density -- 1.7 Local integration by parts formulas and local densities -- 1.8 Random variables -- Chapter 2 Construction of integration by parts formulas -- 2.1 Construction of integration by parts formulas -- 2.1.1 Derivative operators -- 2.1.2 Duality and integration by parts formulas -- 2.1.3 Estimation of the weights -- Iterated derivative operators, Sobolev norms -- Estimate of |?(F)|l -- Bounds for the weights Hq? (F,G) -- 2.1.4 Norms and weights -- 2.2 Short introduction to Malliavin calculus -- 2.2.1 Differential operators -- Step 1: Finite-dimensional di erential calculus in dimension n -- Step 2: Finite-dimensional di erential calculus in arbitrary dimension -- Step 3: Infinite-dimensional calculus -- 2.2.2 Computation rules and integration by parts formulas -- 2.3 Representation and estimates for the density -- 2.4 Comparisons between density functions -- 2.4.1 Localized representation formulas for the density -- 2.4.2 The distance between density functions -- 2.5 Convergence in total variation for a sequence of Wiener functionals -- Chapter 3 Regularity of probability laws by using an interpolation method -- 3.1 Notations -- 3.2 Criterion for the regularity of a probability law -- 3.3 Random variables and integration by parts -- 3.4 Examples -- 3.4.1 Path dependent SDE's. 327 $a3.4.2 Diffusion processes -- 3.4.3 Stochastic heat equation -- 3.5 Appendix A: Hermite expansions and density estimates -- 3.6 Appendix B: Interpolation spaces -- 3.7 Appendix C: Superkernels -- Bibliography -- Part II Functional Ito? Calculus and Functional Kolmogorov Equations -- Preface -- Chapter 4 Overview -- 4.1 Functional Ito? Calculus -- 4.2 Martingale representation formulas -- 4.3 Functional Kolmogorov equations and path dependent PDEs -- 4.4 Outline -- Notations -- Chapter 5 Pathwise calculus for non-anticipative functionals -- 5.1 Non-anticipative functionals -- 5.2 Horizontal and vertical derivatives -- 5.2.1 Horizontal derivative -- 5.2.2 Vertical derivative -- 5.2.3 Regular functionals -- 5.3 Pathwise integration and functional change of variable formula -- 5.3.1 Quadratic variation of a path along a sequence of partitions -- 5.3.2 Functional change of variable formula -- 5.3.3 Pathwise integration for paths of finite quadratic variation -- 5.4 Functionals defined on continuous paths -- 5.5 Application to functionals of stochastic processes -- Chapter 6 The functional Ito? formula -- 6.1 Semimartingales and quadratic variation -- 6.2 The functional Ito? formula -- 6.3 Functionals with dependence on quadratic variation -- Chapter 7 Weak functional calculus for square-integrable processes -- 7.1 Vertical derivative of an adapted process -- 7.2 Martingale representation formula -- 7.3 Weak derivative for square integrable functionals -- 7.4 Relation with the Malliavin derivative -- 7.5 Extension to semimartingales -- 7.6 Changing the reference martingale -- 7.7 Forward-Backward SDEs -- Chapter 8 Functional Kolmogorov equations -- 8.1 Functional Kolmogorov equations and harmonic functionals -- 8.1.1 Stochastic differential equations with path dependent coefficients -- 8.1.2 Local martingales and harmonic functionals. 327 $a8.1.3 Sub-solutions and super-solutions -- 8.1.4 Comparison principle and uniqueness -- 8.1.5 Feynman-Kac formula for path dependent functionals -- 8.2 FBSDEs and semilinear functional PDEs -- 8.3 Non-Markovian stochastic control and path dependent HJB equations -- 8.4 Weak solutions -- Comments and references -- Bibliography. 330 $aThis volume contains lecture notes from the courses given by Vlad Bally and Rama Cont at the Barcelona Summer School on Stochastic Analysis (July 2012). The notes of the course by Vlad Bally, co-authored with Lucia Caramellino, develop integration by parts formulas in an abstract setting, extending Malliavin's work on abstract Wiener spaces. The results are applied to prove absolute continuity and regularity results of the density for a broad class of random processes. Rama Cont's notes provide an introduction to the Functional Itô Calculus, a non-anticipative functional calculus that extends the classical Itô calculus to path-dependent functionals of stochastic processes. This calculus leads to a new class of path-dependent partial differential equations, termed Functional Kolmogorov Equations, which arise in the study of martingales and forward-backward stochastic differential equations. This book will appeal to both young and senior researchers in probability and stochastic processes, as well as to practitioners in mathematical finance. 410 0$aAdvanced Courses in Mathematics - CRM Barcelona,$x2297-0304 606 $aProbabilities 606 $aDifferential equations 606 $aDifferential equations, Partial 606 $aProbability Theory and Stochastic Processes$3https://scigraph.springernature.com/ontologies/product-market-codes/M27004 606 $aOrdinary Differential Equations$3https://scigraph.springernature.com/ontologies/product-market-codes/M12147 606 $aPartial Differential Equations$3https://scigraph.springernature.com/ontologies/product-market-codes/M12155 615 0$aProbabilities. 615 0$aDifferential equations. 615 0$aDifferential equations, Partial. 615 14$aProbability Theory and Stochastic Processes. 615 24$aOrdinary Differential Equations. 615 24$aPartial Differential Equations. 676 $a510 700 $aBally$b Vlad$4aut$4http://id.loc.gov/vocabulary/relators/aut$0756088 702 $aCaramellino$b Lucia$4aut$4http://id.loc.gov/vocabulary/relators/aut 702 $aCont$b Rama$4aut$4http://id.loc.gov/vocabulary/relators/aut 702 $aUtzet$b Frederic$4edt$4http://id.loc.gov/vocabulary/relators/edt 702 $aVives$b Josep$4edt$4http://id.loc.gov/vocabulary/relators/edt 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910254067503321 996 $aStochastic Integration by Parts and Functional Itô Calculus$92004334 997 $aUNINA