The Malliavin CalculusDover Publications, 7 de abr. de 2006 - 128 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. |
Conteúdo
Background material | 7 |
The functional analytic approach | 16 |
The variational approach | 34 |
Direitos autorais | |
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A₁ absolutely continuous abstract Wiener space Applying argument Banach space bilinear Borel measure Brownian motion C¹ function chapter continuous with respect defined Definition denote density differentiable with respect distribution dy(w dy(x exists a constant finite following result Furthermore Gaussian measure Girsanov theorem gives GL(d Hence Hilbert space Hörmander's condition Hörmander's theorem implies inequality infinite dimensional inner product iterating Itô Itô's lemma kt/m L²(y Lebesgue measure Lipschitz Malliavin calculus matrix measure on Rd measures induced n-dimensional Brownian motion norm obtained orthonormal basis polynomial proof of Theorem prove quasi-invariant random variable real-valued sequence smooth solution stochastic differential equation stochastic integral subspace Suppose test function Theorem 1.9(iv Theorem 2.5 theory vector fields w₁ Wiener measure wx(t y₁