Algorithmes stochastiques pour la gestion du risque et l'indexation de bases de données de média

Victor Reutenauer 1
1 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
Abstract : This thesis proposes different problems of stochastic control and optimization that can be solved only thanks approximation. On one hand, we develop methodology aiming to reduce or suppress approximations to obtain more accurate solutions or something exact ones. On another hand we develop new approximation methodology in order to solve quicker larger scale problems. We study numerical methodology to simulated differential equations and enhancement of computation of expectations. We develop quantization methodology to build control variate and gradient stochastic methods to solve stochastic control problems. We are also interested in clustering methods linked to quantization, and principal composant analysis or compression of data thanks neural networks. We study problems motivated by mathematical finance, like stochastic control for the hedging of derivatives in incomplete market but also to manage huge databases of media commonly known as big Data in chapter 5. Theoretically we propose some upper bound for convergence of the numerical method used. This is the case of optimal hedging in incomplete market in chapter 3 but also an extension of Beskos-Roberts methods of exact simulation of stochastic differential equations in chapter 4. We present an original application of karhunen-Loève decomposition for a control variate of computation of expectation in chapter 2.
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Victor Reutenauer. Algorithmes stochastiques pour la gestion du risque et l'indexation de bases de données de média. Mathématiques générales [math.GM]. Université Côte d'Azur, 2017. Français. ⟨NNT : 2017AZUR4018⟩. ⟨tel-01544854⟩

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