p-variations approchées et erreurs d'arrondis

Abstract : In this thesis, we study the asymptotic properties of processes observed discretely in time, and with a round-off error, when both the time lag between two observations and the size of the round-off error go to 0. This thesis is divided into three parts; in the first one we recall some elements about semimartingale and the different kinds of convergences we will use. Chapter 2 is devoted to the study of p,q-variations for a 2-dimensional Brownian motion ; we find that two important parameters appear: the first is the quotient between the size of the round-off and the square root of the time lag and the second the covariance matrix associated with the brownian motion. We obtain different laws of large numbers, that depend whether the first parameter converges or diverges and whether this matrix is or is not invertible. When it is not, very different behaviours arise when the two components of the Brownian motion have a rational quotient and when it is irrational. In chapter 3 we deal with asymptotic p-variations of rounded-off semimartingales. We first prove laws of large numbers for either renormalized or non-renormalized p-variations, as well as a law of large numbers for 2-dimensional continuous semimartingales, using some results from chapter 2. In some cases, namely in the non-renormalized one, we also prove, with some unavoidable further assumptions on the semimartingale, a central limit theorem.
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Submitted on : Wednesday, June 15, 2011 - 1:37:59 PM
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Pierre-Henri Cumenge. p-variations approchées et erreurs d'arrondis. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2011. Français. ⟨tel-00600623⟩

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