Statistical inference of Ornstein-Uhlenbeck processes : generation of stochastic graphs, sparsity, applications in finance

Abstract : The subject if this thesis is the statistical inference of multi-dimensional Ornstein-Uhlenbeck processes. In a first part, we introduce a model of stochastic graphs, defined as binary observations of a trajectory. We show then that it is possible to retrieve the dynamic of the underlying trajectory from the binary observations. For this, we build statistics of the stochastic graph and prove new results on their convergence in the long-time, high-frequency setting. We also analyse the properties of the stochastic graph from the point of view of evolving networks. In a second part, we work in the setting of complete information and continuous time. We add then a sparsity assumption applied to the drift matrix coefficient of the Ornstein-Uhlenbeck process. We prove sharp oracle inequalities for the Lasso estimator, construct a lower bound on the estimation error for sparse estimators and show optimality properties of the Adaptive Lasso estimator. Then, we apply the methods to estimate mean-return properties of real-world financial datasets: daily returns of SP500 components and EURO STOXX 50 Dividend Future prices.
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Gustaw Matulewicz. Statistical inference of Ornstein-Uhlenbeck processes : generation of stochastic graphs, sparsity, applications in finance. Statistics [math.ST]. Université Paris-Saclay, 2017. English. ⟨NNT : 2017SACLX066⟩. ⟨tel-01688026⟩

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