NDLI: Sufficient conditions for the invertibility of adapted perturbations of identity on the Wiener space

Content Provider	Springer Nature Link
Author	Üstünel, Ali Süleyman Zakai, Moshe
Copyright Year	2006
Abstract	Let (W, H, μ) be the classical Wiener space. Assume that U = I W + u is an adapted perturbation of identity, i.e., u : W → H is adapted to the canonical filtration of W. We give some sufficient analytic conditions on u which imply the invertibility of the map U. In particular it is shown that if $${u\in {\rm ID}_{p,1}(H)}$$ is adapted and if $${\exp(\frac{1}{2}\\|\nabla u\\|_2^2-\delta u)\in L^q(\mu)}$$ , where p −1 + q −1 = 1, then I W + u is almost surely invertible. With the help of this result it is shown that if $${\nabla u\in L^\infty(\mu,H\otimes H)}$$ , then the Girsanov exponential of u times the Wiener measure satisfies the logarithmic Sobolev inequality and this implies the invertibility of U = I W + u . As a consequence, if, there exists an integer k ≥ 1 such that $${\\|\nabla^k u\\|_{H^{\otimes(k+1)}}\in L^\infty(\mu)}$$ , then I W + u is again almost surely invertible under the almost sure continuity hypothesis of $${t\to\nabla^i \dot{u}_t}$$ for i ≤ k − 1.
Ending Page	234
Page Count	28
Starting Page	207
File Format	PDF
ISSN	01788051
e-ISSN	14322064
Journal	Probability Theory and Related Fields
Issue Number	1
Volume Number	139
Language	English
Publisher	Springer-Verlag
Publisher Date	2007-01-06
Publisher Place	Berlin/Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Logarithmic sobolev inequality Measures and integration on abstract linear spaces Variational problems in a geometric measure-theoretic setting Adapted Mathematical and Computational Physics Probability Theory and Stochastic Processes Lipschitz (HÃ¶lder) classes Stochastic calculus of variations and the Malliavin calculus H-C1 maps Stochastic integrals Random operators and equations Perturbation of identity Multiplicity Operations Research/Decision Theory Applications in optimization, convex analysis, mathematical programming, economics Mathematical Biology in General Carleman’s inequality Monotone operators and generalizations Geometric theory, characteristics, transformations Wiener space Continuity and singularity of induced measures Signal detection and filtering Nonlinear elliptic equations Quantitative Finance Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) Smoothness and regularity of solutions Change of variables formula Statistics for Business/Economics/Mathematical Finance/Insurance Gaussian processes Wiener measure Malliavin calculus Applications to minimal surfaces (problems in two independent variables)
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Analysis Statistics, Probability and Uncertainty

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