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Equations aux dérivées partielles en finance : problèmes inverses et calibration de modèle.

Abstract : In the first part of this thesis, we studied the impact on prices of options volatility estimation errors. In diffusion models used ennance, a diffusion coefficient fonctinnelle (:,:) modeled the volatility of an asset Financial. This coefficient is estimated from observations has thus tainting of statistical errors.This leads to a problem of transition to the limit (homogenization) in parabolic equations with coefficients random. In this work we obtained estimates of the speed of local convergence on the solution of a PDE parabolic random coefficients, when the diffusion coefficient is a field random converging to a limit function. This result allows to study the im- pact on prices of options volatility estimation errors into different cases degures. This method is applied to evaluate the uncertainty on the options has Barrier-diffusion models when the volatility reconstructs the Dupire formula from data on discrete option prices. The second part of this thesis concerns the study of inverse problems for certain class of evolution equations int? Egro-differential occurring in the study Evaluation models bases on Levy processes. We studied an approach to these inverse problems by Tikhonov regularization. This approach allows stably to reconstruct the parameters of a Markov model with a jump from the observation of a number or options. Chapter 4 discusses the theoretical foundations of this approach and proposes a parametrization Levy measures by the square root of the density, thereby reducing the problem in a Hilbert space. The Tikhonov regularization is proposed to minimize the squared deviation from the observed price? are more a Hilbert norm parameters. Results of existence, stability and convergence of the regularized solution of the problem are then obtained under fairly general assumptions of, additional hypotheses (source conditions) can obtain an estimate of the speed of convergence. The choice of the regularization parameter, delicate subject, the subject of a detailed discussion. Chapter 5 provides a numerical algorithm for calculating the regularized solution of the problem and study of the Performance of this algorithm in different models with jumps. The algorithm is based on the use of a gradient algorithm for minimizing the regularized functional: the gradient is computed by solving an integrodifferential equation with source term (equation Assistant). This work generalizes those of Lagnado & Osher, and Egger & Crépey Engl case of integrodifferential equations. The numerical tests show that this algorithm allows to build a stably Levy process has a set of class
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Submitted on : Thursday, July 22, 2010 - 11:12:41 AM
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  • HAL Id : pastel-00003888, version 1



Moeiz Rouis. Equations aux dérivées partielles en finance : problèmes inverses et calibration de modèle.. Mathématiques [math]. Ecole Polytechnique X, 2007. Français. ⟨pastel-00003888⟩



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