1,721,053 research outputs found
Kusuoka-Stroock type bounds for densities related to low-dimensional projections of high-dimensional SDE
Get PDFOne of the purposes of this thesis is to use Malliavin calculus and
Stochastic Taylor expansions to study the densities of interacting
systems of stochastic differential equations (SDE), seen as
projections of SDE onto a low-dimensional space, and to control the
dependence of the constants on the dimension of the background
space. The setting includes time-dependent SDE and a relatively large
class of path-dependent SDE. Several results also shed light on the
classical theory of SDE, independently of the control on the
constants.
In Part 1, assuming the system satisfies suitably defined projected
equivalents of the classic ellipticity or weak Hörmander conditions,
we prove Gaussian estimates in terms of the Euclidean distance where,
provided natural assumptions, for a fixed target-space
dimension, the constants depend polynomially on the background dimension, and, in the elliptic case, on the number of
driving Brownian motions.
In Part 2, we first define suitable generalisations of
(time-dependent) control distances and prove Kusuoka-Stroock type
results without control on the constants.
In particular, we obtain a time-dependent extension of a result of
Léandre about SDE with non-trivial drifts, i.e., drifts which are not
uniformly contained in the span of the other vector fields.
Then, we introduce a condition which we call the `Progressive
Hörmander condition' and prove similar control-type estimates valid
under this assumption, with polynomial control on the growth
of the constants with background space dimension. The condition is of independent interest in the
study of SDE, and shows the connection between the classic
works of Ben Arous, Kusuoka, Léandre and Stroock, and the more recent
works of Bally, Caramellino, Delarue, Menozzi and Pigato. To main
technique required is the study of density and scaling properties of
some careful choice of linear combinations of terms of the signature
of the driving path.
In Part 3, we introduce a stricter condition called the `separated
progressive Hörmander condition', and prove lower bounds and local
strict positivity under this assumption. (By `local' we mean local
around the solution of the deterministic ODE driven by a null control,
rather than local round the initial point.) The main technical
difficulty is the identification of points contained in the interior
of the support of the log-signature of the path in
the d dimensional Euclidean space composed of d Brownian motions and a deterministic linear
component.
The purpose of Part 4 is to use some results and techniques of the
rest of the thesis to prove extensions of a theorem of Löcherbach
about uniformly elliptic interacting branching diffusions
Geometry and Stochastic Calculus on Wasserstein spaces
Get PDFThe main object of interest in this thesis is P(M) – the space of probability measures on a manifold endowed with the Wasserstein distance:
In chapter 1 we give the most basic topological facts and introduce a locally convex topology on P∞ (the space of smooth positive densities) to identify this space as infinite dimensional manifold.
In chapter 2 we develop further the Riemannian calculus on P resp. P∞ where the different approaches (calculus of variation, Riemannian geometry on spaces of smooth mappings) are shown to be equivalent on P∞ .
In chapter 3 we restrict ourself tomeasures on the unit circle and give calculations of renormalized Laplacians on the respective Wasserstein spaces, seen as the Hilbert-Schmidt trace of the Hessian: This trace depends on a real parameter s and has an analytic continuation as a function of s ∈ C \ {1} which enables us to calculate evaluate at s = 0: The square-field operator of the Wassersein Laplacian equals the squared Wasserstein gradient times the volume of the unit circle.
In chapter 4 we give an approximation of the Wasserstein space P ([0, 1]) by spaces of box-type measures which are geodesically convex and can be mapped isometrically via a mapping simplex , where a sticky diffusion process is constructed. We show that image of this processes constitute a tight family in C(R_+P ([0, 1])) with respect to the Skorohod topology.
In the last chapter we restrict ourselves to the space of histograms on the unit interval. We calculate the Wasserstein distances numerically and obtain a Riemannian metric on the simplex. We investigate explosion behaviour of the respective diffusion processes in dimension 1 and 2
Curvature-Dimension Bounds and Functional Inequalities : Localization, Tensorization and Stability
Get PDFThis work is devoted to the analysis of abstract metric measure spaces (M,d,m) satisfying the curvature-dimension condition CD(K,N) presented by Sturm and in a similar form by Lott and Villani. In the first part, we introduce the notion of a Borell-Brascamp-Lieb inequality in the setting of metric measure spaces denoted by BBL(K,N). This inequality holds true on metric measure spaces fulfilling the curvature-dimension condition CD(K,N) and is stable under convergence of metric measure spaces with respect to the transportation distance. In the second part, we prove that the local version of CD(K,N) is equivalent to a global condition CD*(K,N), slightly weaker than the usual global one. This so-called reduced curvature-dimension condition CD*(K,N) has the localization property. Furthermore, we show its stability and the tensorization property. As an application we conclude that the fundamental group of a metric measure space (M,d,m) is finite whenever it satisfies locally the curvature-dimension condition CD(K,N) with positive K and finite N. In the third part, we study cones over metric measure spaces. We deduce that the n-Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-dimension condition CD(0,n+1) and that the n-spherical cone over the same manifold fulfills CD(n,n+1)
CONTRIBUTIONS TO THE STATISTICS OF RANDOM PROCESSES USING MALLIAVIN CALCULUS
No full textIn this dissertation we present several applications of Malliavin calculus, both to the statistical analysis of continuous time stochastic processes and to limit theorems for non-linear functionals of Gaussian Fields. Malliavin calculus extends techniques of classical calculus of variations from deterministic functions to random variables. In Malliavin calculus, the so called Malliavin derivative and its adjoint, the divergence operator, are combined with the theory of Hilbert spaces. Just as classical calculus, this theory has proved to be a powerful tool and its applications vary from the existence of densities, to the construction of estimators and the study of weak convergence of sequences of random variables and random vectors, with a special focus on normal approximations.
The first part of the present document is essentially a generalization of a result of Privault and Réveillac (2008), which extends a seminal paper of Stein (1956). Stein has shown that, under certain conditions, there are biased estimators which perform better than the standard estimator for the mean of a multivariate normal vector. It has been shown by Privault and Réveillac that a similar statement holds for Gaussian processes and we shall present a generalization of their work to continuous time models, where the noise is either a chaotic Brownian martingale or a non-martingale noise living in the second Wiener chaos. This first part of the work corresponds to the paper "Drift estimation with non-gaussian noise using Malliavin Calculus" (2015) which has been published by the Electronic Journal of Statistics.
In the second part of the work we give necessary and sufficient criteria for the convergence of sequences of random variables, living in a fixed sum of Wiener chaoses, to a limit which lives in the sum of the first two Wiener chaoses. Our results extend the important findings of Nualart and Peccati (2005), the so-called Fourth Moment Theorem, and a recent finding of Azmoodeh, Peccati and Poly (2014). Our criteria make use of the so-called Gamma-operators which are derived from scalar products of Malliavin derivatives and the infinitesimal generator of the Ornstein-Uhlenbeck semi-group, see for instance Azmoodeh, Peccati and Poly (2014). This part corresponds to the paper "Weak convergence on Wiener space: targeting the first two chaoses" (2017) which has been submitted to the Latin American Journal of Probability and Mathematical Statistics (ALEA).
In the last part of the present work we consider a sequence living in a fixed Wiener chaos and converging in law to a normal variable. A second sequence is supposed to converge in law to a target variable which is the sum of a linear combination of independent chi-square distributed random variables and an independent normal variable. We derive conditions under which the sequence of random vectors, formed by both sequences of random variables, converges in law. We use again Gamma-operators and cumulants to derive necessary and sufficient conditions which can be seen as generalization of results of Peccati and Tudor (2005) for Gaussian limits in the case of sequences of random vectors which converge componentwise. We apply methods developed by Nourdin and Peccati (2009) to examine the rate of convergence of a sequence of double Wiener integrals towards a normal variable
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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