Le sujet principal de cette thèse est de proposer de nouvelles méthodes d'extractions de signaux avec applications en finance. Par signaux, nous entendons soit un signal sur lequel repose une stratégie d'investissement; soit un signal perturbé par un bruit, que nous souhaitons retrouver. Ainsi, la première partie de la thèse étudie la contagion en volatilité historique autours des annonces de résultats des entreprises du Nasdaq. Nous trouvons qu'autours de l'annonce, l'entreprise reportant ses résultats, génère une contagion persistante en volatilité à l’encontre des entreprises appartenant au même secteur. Par ailleurs, nous trouvons que la contagion en volatilité varie, selon le type de nouvelles reportées, l'effet de surprise, ou encore par le sentiment de marché à l'égard de l'annonceur. La deuxième partie de cette thèse adapte des techniques de dé-bruitage venant de l'imagerie, à des formes de bruits présentent en finance. Ainsi, un premier article, co-écrit avec Matthieu Garcin, propose une technique de dé-bruitage innovante, permettant de retrouver un signal perturbé par un bruit à variance non-constante. Cet algorithme est appliqué en finance à la modélisation de la volatilité. Un second travail s'intéresse au dé-bruitage d'un signal perturbé par un bruit asymétrique et leptokurtique. En effet, nous adaptons un modèle de Maximum A Posteriori, couramment employé en imagerie, à des bruits suivant des lois de probabilité de Student, Gaussienne asymétrique et Student asymétrique. Cet algorithme est appliqué au dé-bruitage de prix d'actions haute-fréquences. L'objectif étant d'appliquer un algorithme de reconnaissance de formes sur les extrema locaux du signal dé-bruité. / The main objective of this PhD dissertation is to set up new signal extraction techniques with applications in Finance. In our setting, a signal is defined in two ways. In the framework of investement strategies, a signal is a function which generates buy/sell orders. In denoising theory, a signal, is a function disrupted by some noise, that we want to recover. A first part of this PhD studies historical volatility spillovers around corporate earning announcements. Notably, we study whether a move by one point in the announcer historical volatility in time t will generate a move by beta percents in time t+1. We find evidences of volatility spillovers and we study their intensity across variables such as : the announcement outcome, the surprise effect, the announcer capitalization, the market sentiment regarding the announcer, and other variables. We illustrate our finding by a volatility arbitrage strategy. The second part of the dissertation adapts denoising techniques coming from imagery : wavelets and total variation methods, to forms of noise observed in finance. A first paper proposes an denoising algorithm for a signal disrupted by a noise with a spatially varying standard-deviation. A financial application to volatility modelling is proposed. A second paper adapts the Bayesian representation of the Rudin, Osher and Fatemi approach to asymmetric and leptokurtic noises. A financial application is proposed to the denoising of intra-day stock prices in order to implement a pattern recognition trading strategy.

In this thesis we study novel higher order total variation-based variational methods for digital image reconstruction. These methods are formulated in the context of Tikhonov regularisation. We focus on regularisation techniques in which the regulariser incorporates second order derivatives or a sophisticated combination of first and second order derivatives. The introduction of higher order derivatives in the regularisation process has been shown to be an advantage over the classical first order case, i.e., total variation regularisation, as classical artifacts such as the staircasing effect are significantly reduced or totally eliminated. Also in image inpainting the introduction of higher order derivatives in the regulariser turns out to be crucial to achieve interpolation across large gaps. First, we introduce, analyse and implement a combined first and second order regularisation method with applications in image denoising, deblurring and inpainting. The method, numerically realised by the split Bregman algorithm, is computationally efficient and capable of giving comparable results with total generalised variation (TGV), a state of the art higher order method. An additional experimental analysis is performed for image inpainting and an online demo is provided on the IPOL website (Image Processing Online). We also compute and study properties of exact solutions of the one dimensional total generalised variation problem with L^{2} data fitting term, for simple piecewise affine data functions, with or without jumps . This gives an insight on how this type of regularisation behaves and unravels the role of the TGV parameters. Finally, we introduce, study and analyse a novel non-local Hessian functional. We prove localisations of the non-local Hessian to the local analogue in several topologies and our analysis results in derivative-free characterisations of higher order Sobolev and BV spaces. An alternative formulation of a non-local Hessian functional is also introduced which is able to produce piecewise affine reconstructions in image denoising, outperforming TGV.