## Stochastic Integration by Parts and Functional Itô CalculusThis 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. |

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Stochastic Integration by Parts and Functional Itô Calculus Vlad Bally,Lucia Caramellino,Rama Cont No preview available - 2016 |

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absolutely continuous Applying Bally bounded Brownian motion Cº(X conditional expectation consider construction convergence define Definition denote density dt x d eaſists estimates F-adapted FBSDE finite quadratic variation finite-dimensional functional F Functional Itô Calculus functional Itô formula Functional Kolmogorov Equation Hölder continuous Hölder's inequality horizontal inequality integrands Itô Calculus Itô formula law of F Lebesgue measure left-continuous Lemma Let F LP(Q Malliavin calculus Malliavin derivative Moreover multiinder multiindex non-anticipative functional norm notation obtain path dependent PDEs probability measure Proposition prove quadratic variation random variables regularity representation formula respect semimartingale sequence of partitions simple functionals Sº(X Sobolev spaces square integrable stochastic differential equation stochastic integral stopped path supp(X TF(t unique universal constant vertical derivative weak solution Wiener process Wiener space Wºº