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Penalized estimators for non linear inverse problems

Jean-Michel Loubes, Carenne Ludeña (2010)

ESAIM: Probability and Statistics

In this article we tackle the problem of inverse non linear ill-posed problems from a statistical point of view. We discuss the problem of estimating an indirectly observed function, without prior knowledge of its regularity, based on noisy observations. For this we consider two approaches: one based on the Tikhonov regularization procedure, and another one based on model selection methods for both ordered and non ordered subsets. In each case we prove consistency of the estimators and show...

Properties of local-nondeterminism of Gaussian and stable random fields and their applications

Yimin Xiao (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

In this survey, we first review various forms of local nondeterminism and sectorial local nondeterminism of Gaussian and stable random fields. Then we give sufficient conditions for Gaussian random fields with stationary increments to be strongly locally nondeterministic (SLND). Finally, we show some applications of SLND in studying sample path properties of ( N , d ) -Gaussian random fields. The class of random fields to which the results are applicable includes fractional Brownian motion, the Brownian...

Quelques espaces fonctionnels associés à des processus gaussiens

Z. Ciesielski, G. Kerkyacharian, B. Roynette (1993)

Studia Mathematica

The first part of the paper presents results on Gaussian measures supported by general Banach sequence spaces and by particular spaces of Besov-Orlicz type. In the second part, a new constructive isomorphism between the just mentioned sequence spaces and corresponding function spaces is established. Consequently, some results on the support function spaces for the Gaussian measure corresponding to the fractional Brownian motion are proved. Next, an application to stochastic equations is given. The...

Régularité Besov des trajectoires du processus intégral de Skorokhod

Gérard Lorang (1996)

Studia Mathematica

Let W t : 0 ≤ t ≤ 1 be a linear Brownian motion, starting from 0, defined on the canonical probability space (Ω,ℱ,P). Consider a process u t : 0 ≤ t ≤ 1 belonging to the space ℒ 2 , 1 (see Definition II.2). The Skorokhod integral U t = ʃ 0 t u δ W is then well defined, for every t ∈ [0,1]. In this paper, we study the Besov regularity of the Skorokhod integral process t ↦ U t . More precisely, we prove the following THEOREM III.1. (1)If 0 < α < 1/2 and u ∈ ℒ p , 1 with 1/α < p < ∞, then a.s. t ↦ U t ∈ ℬ p , q α for all q ∈ [1,∞], and t → U t ∈ ℬ p , ∞ α , 0 . (2) For every even integer p ≥...

Regularity of Gaussian white noise on the d-dimensional torus

Mark C. Veraar (2011)

Banach Center Publications

In this paper we prove that a Gaussian white noise on the d-dimensional torus has paths in the Besov spaces B p , ∞ - d / 2 ( d ) with p ∈ [1,∞). This result is shown to be optimal in several ways. We also show that Gaussian white noise on the d-dimensional torus has paths in the Fourier-Besov space b ̂ p , ∞ - d / p ( d ) . This is shown to be optimal as well.

Reinsurance-a new approach

Adam Paszkiewicz, Jakub Olejnik (2010)

Banach Center Publications

We describe a new model of multiple reinsurance. The main idea is that the reinsurance premium is paid conditionally. It is motivated by some analysis of the ultimate price of the reinsurance contract. For simplicity we assume that the underlying risk pricing functional is the L₂-norm. An unexpected relation to the general theory of sample regularity of stochastic processes is given.

Revisiting the sample path of Brownian motion

S. James Taylor (2006)

Banach Center Publications

Brownian motion is the most studied of all stochastic processes; it is also the basis for stochastic analysis developed in the second half of the 20th century. The fine properties of the sample path of a Brownian motion have been carefully studied, starting with the fundamental work of Paul Lévy who also considered more general processes with independent increments and extended the Brownian motion results to this class. Lévy showed that a Brownian path in d (d ≥ 2) dimensions had zero Lebesgue measure;...

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