The Malliavin Calculus
Dover Publications, 2006 - 113 páginas
This introduction to Malliavin's stochastic calculus of variations is suitable for graduate students and professional mathematicians. Author Denis R. Bell particularly emphasizes the problem that motivated the subject's development, with detailed accounts of the different forms of the theory developed by Stroock and Bismut, discussions of the relationship between these two approaches, and descriptions of a variety of applications.
The first chapter covers enough technical background to make the subsequent material accessible to readers without specialized knowledge of stochastic analysis. Succeeding chapters examine the functional analytic and variational approaches (with extensive explorations of the work of Stroock and Bismut); and elementary derivation of Malliavin's inequalities and a discussion of the different forms of the theory; and the non-degeneracy of the covariance matrix under Hormander's condition. The text concludes with a brief survey of applications of the Malliavin calculus to problems other than Hormander's.
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A discussion of the dilferent forms of the theory
Nondegeneracy of the covariance matrix lmder H6rmanders
Some further applications of the Malliavin calculus
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absolutely continuous abstract Wiener space argument Banach space bilinear Borel measure bounded ﬁrst chapter condition continuous with respect converges covariance matrix deﬁned Deﬁnition 2.1 denote differentiable with respect diﬁerential domain dY(w elementary estimates exists a constant ﬁnite dimensional ﬁrst and second ﬁxed following result follows from Lemma Furthermore Gaussian measure Girsanov theorem gives GL(d Hence Hilbert space hypotheses implies independent inequality inﬁnite dimensional inner product introduce invertible iterating Ito’s lemma Lebesgue measure linear operator Lipschitz Malliavin calculus MATHEMATICAL measures induced n-dimensional Brownian motion non-degeneracy norm notation Note obtained orthonormal basis polynomial problem proof of Theorem prove quasi-invariant random variable real-valued satisﬁes the equation section 3.2 sequence shown solution stochastic differential equation stochastic integral Stratonovich Stroock and Bismut subspace suﬂicient Suppose test function Theorem 1.9(iv Theorem 2.5 vector ﬁelds Wiener measure