Approximation du temps local et intégration par régularisation

Abstract : The setting of this work is the integration by regularization of Russo and Vallois. The first part studies schemes of approximation of the local time of continuous semimartingales. It is proven that, if $X$ is a reversible diffusion, then $\frac{1}{\epsilon}\int_0^t \left( \indi_{\{ y < X_{s+\epsilon}\}} - \indi_{\{ y < X_{s}\}} \right) \left( X_{s+\epsilon}-X_{s} \right)ds$ converges to $L_t^y(X)$, in probability uniformly on the compact sets, when $\epsilon \to 0$. From this first schema, two other schemas of approximation for the local time are found. One converges in the semi-martingale case, the other in the Brownian case. Moreover, in the Brownian case, we estimate the rate of convergence in $L^2(\Omega)$ and a result of almost sure convergence is proven. The second part study the forward integral and the generalized quadratic variation, which have been defined by convergence of families of integrals, in probability uniformly on the compacts sets. In the case of Hölder processes, the almost sure convergence is proven. Finally, the second order convergence is studied in many cases.
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https://tel.archives-ouvertes.fr/tel-01748283
Contributor : Blandine Berard Bergery <>
Submitted on : Wednesday, October 24, 2007 - 2:58:29 PM
Last modification on : Thursday, February 25, 2021 - 10:54:06 AM
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• HAL Id : tel-01748283, version 2

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Blandine Berard Bergery. Approximation du temps local et intégration par régularisation. Mathématiques [math]. Université Henri Poincaré - Nancy 1, 2007. Français. ⟨NNT : 2007NAN10058⟩. ⟨tel-01748283v2⟩

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