122 research outputs found
An essay on the general theory of stochastic processes
This text is a survey of the general theory of stochastic processes, with a
view towards random times and enlargements of filtrations. The first five
chapters present standard materials, which were developed by the French
probability school and which are usually written in French. The material
presented in the last three chapters is less standard and takes into account
some recent developments.Comment: Published at http://dx.doi.org/10.1214/154957806000000104 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Enlargements of filtrations and path decompositions at non-stopping times
Az\'{e}ma associated with an honest time L the supermartingale
and established some of its
important properties. This supermartingale plays a central role in the general
theory of stochastic processes and in particular in the theory of progressive
enlargements of filtrations. In this paper, we shall give an additive
characterization for these supermartingales, which in turn will naturally
provide many examples of enlargements of filtrations. In particular, we use
this characterization to establish some path decomposition results, closely
related to or reminiscent of Williams' path decomposition results.Comment: New titlle for this second version; Typos corrected; same as the
published version in Prob. Theory and Related Fields 136 (4), 2006, 524-54
A class of remarkable submartingales
In this paper, we consider the special class of positive local submartingales
(X_{t}) of the form: X_{t}=N_{t}+A_{t}, where the measure (dA_{t}) is carried
by the set {t: X_{t}=0}. We show that many examples of stochastic processes
studied in the literature are in this class and propose a unified approach
based on martingale techniques to study them. In particular, we establish some
martingale characterizations for these processes and compute explicitly some
distributions involving the pair (X_{t},A_{t}). We also associate with X a
solution to the Skorokhod's stopping problem for probability measures on the
positive half-line.Comment: Typos corrected. Close to the published versio
The distribution of eigenvalues of randomized permutation matrices
In this article we study in detail a family of random matrix ensembles which
are obtained from random permutations matrices (chosen at random according to
the Ewens measure of parameter ) by replacing the entries equal to
one by more general non-vanishing complex random variables. For these
ensembles, in contrast with more classical models as the Gaussian Unitary
Ensemble, or the Circular Unitary Ensemble, the eigenvalues can be very
explicitly computed by using the cycle structure of the permutations. Moreover,
by using the so-called virtual permutations, first introduced by Kerov,
Olshanski and Vershik, and studied with a probabilistic point of view by
Tsilevich, we are able to define, on the same probability space, a model for
each dimension greater than or equal to one, which gives a meaning to the
notion of almost sure convergence when the dimension tends to infinity. In the
present paper, depending on the precise model which is considered, we obtain a
number of different results of convergence for the point measure of the
eigenvalues, some of these results giving a strong convergence, which is not
common in random matrix theory
A definition and some characteristic properties of pseudo-stopping times
Recently, D. Williams \cite{williams} gave an explicit example of a random
time associated with Brownian motion such that is not a
stopping time but for every bounded
martingale . The aim of this paper is to give some characterizations for
such random times, which we call pseudo-stopping times, and to construct
further examples, using techniques of progressive enlargements of filtrations.Comment: 30 pages; to appear in Annals of Probabilit
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