FERMANIAN Jean David

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This thesis can be divided into three parts.In the first part, we study methods of adaptation to the noise level in the high dimensional linear regression model. We prove that two square-root estimators can reach the minimax estimation and prediction speeds. We show that a similar version built from mean medians can still reach the same optimal speeds and is robust to the presence of outliers.The second part is devoted to the analysis of several conditional dependence models. We propose several tests of the simplifying hypothesis that a conditional copula is constant with respect to its conditioning event, and we prove the consistency of a semi-parametric resampling technique. If the conditional copula is not constant with respect to its conditioning variable, then it can be modeled via its conditional Kendall's tau. We study the estimation of this conditional dependence parameter under 3 different approaches: kernel techniques, regression type models and classification algorithms.The last part gathers two contributions in the field of inference.We compare and propose different estimators of regular conditional functionals using U-statistics. Finally, we study the construction and theoretical properties of confidence intervals for mean ratios under different choices of assumptions and paradigms.

The extreme joint behavior between random variables is of particular interest in many applications in environmental sciences, finance, insurance or risk management. For example, this behavior plays a central role in the evaluation of natural disaster risks. A misspecification of the dependence between random variables can lead to a dangerous underestimation of the risk, especially at the extreme level. The first objective of this thesis is to develop inference techniques for Archimax copulas. These dependence models can capture any type of asymptotic dependence between the extremes and, simultaneously, model the risks attached to the mean level. An Archimax copula is characterized by its two functional parameters, the stable caudal dependence function and the Archimedean generator that acts as a distortion affecting the extreme dependence regime. Conditions are derived so that the generator and the caudal function are identifiable, so that a semi-parametric inference approach can be developed. Two nonparametric estimators of the caudal function and a moment-based estimator of the generator, assuming that the latter belongs to a parametric family, are advanced. The asymptotic behavior of these estimators is then established under non-restrictive regularity assumptions and the finite sample performance is evaluated through a simulation study. A hierarchical (or "cluster") construction that generalizes the Archimax copulas is proposed in order to provide more flexibility, making it more suitable for practical applications. The extreme behavior of this new dependence model is then studied, which leads to a new way of constructing stable caudal dependence functions. The Archimax copula is then used to analyze the monthly precipitation maxima, observed at three weather stations in Brittany. The model seems to fit the data very well, both for light and heavy precipitation. The non-parametric estimator of the caudal function reveals an extreme asymmetric dependence between stations, reflecting the movement of thunderstorms in the region. An application of the hierarchical Archimax model to a precipitation dataset containing 155 stations is then presented, in which asymptotically dependent groups of stations are determined via a "clustering" algorithm specifically adapted to the model. Finally, possible methods to model inter-cluster dependence are discussed.

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We introduce so-called single-index copulas. They are semi-parametric conditional copulas whose parameter is an unknown link function of a univariate index only. We propose estimates of this link function and of the finite-dimensional unknown parameter. The asymptotic properties of the latter estimates are stated. Thanks to some properties of conditional Kendall's tau, we illustrate our technical conditions with several usual copula families.

This thesis deals with the theory of stochastic equations in finite dimension. In the first part, we derive necessary and sufficient geometric conditions on the coefficients of a stochastic differential equation for the existence of a solution constrained to remain in a closed domain, under weak regularity conditions on the coefficients.In the second part, we address existence and uniqueness problems of stochastic Volterra equations of convolutional type. These equations are in general non-Markovian. We establish their correspondence with infinite-dimensional equations which allows us to approximate them by finite-dimensional Markovian stochastic differential equations. Finally, we illustrate our results by an application in mathematical finance, namely the modeling of rough volatility. In particular, we propose a stochastic volatility model that provides a good compromise between flexibility and tractability.

This thesis deals with dependence problems between stochastic processes in continuous time. In a first part, new copulas are established to model the dependence between two Brownian movements and to control the distribution of their difference. It is shown that the class of admissible copulas for Brownians contains asymmetric copulas. With these copulas, the survival function of the difference of the two Brownians is higher in its positive part than with a Gaussian dependence. The results are applied to the joint modeling of electricity prices and other energy commodities. In a second part, we consider a discretely observed stochastic process defined by the sum of a continuous semi-martingale and a compound Poisson process with mean reversion. An estimation procedure for the mean-reverting parameter is proposed when the mean-reverting parameter is large in a high frequency finite horizon statistical framework. In a third part, we consider a doubly stochastic Poisson process whose stochastic intensity is a function of a continuous semi-martingale. To estimate this function, a local polynomial estimator is used and a window selection method is proposed leading to an oracle inequality. A test is proposed to determine if the intensity function belongs to a certain parametric family. With these results, the dependence between the intensity of electricity price peaks and exogenous factors such as wind generation is modeled.

This paper deals with the high dimensionality problem in multivariate GARCH processes. The author proposes a new vine-GARCH dynamics for correlation processes parameterized by an undirected graph called "vine". This approach directly generates definite-positive matrices and encourages parsimony. After establishing existence and uniqueness results for stationary solutions of the vine-GARCH model, the author analyzes the asymptotic properties of the model. He then proposes a general framework of penalized M-estimators for dependent processes and focuses on the asymptotic properties of the adaptive Sparse Group Lasso estimator. The high dimension is treated by considering the case where the number of parameters diverges with the sample size. The asymptotic results are illustrated by simulated experiments. Finally in this framework the author proposes to generate the sparsity for dynamics of variance-covariance matrices. To do so, the class of multivariate ARCH models is used and the corresponding processes are estimated by penalized ordinary least squares.

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In this thesis, we consider the joint law modeling of order statistics, i.e. random vectors with almost surely ordered components. The first part is dedicated to the probabilistic modeling of order statistics of maximum entropy with fixed marginals. Since the marginals are fixed, the characterization of the joint distribution amounts to considering the associated copula. In Chapter 2, we present an auxiliary result on maximum entropy copulas with fixed diagonal. A necessary and sufficient condition is given for the existence of such a copula, as well as an explicit formula for its density and entropy. The solution of the entropy maximization problem for order statistics with fixed marginals is presented in Chapter 3. Explicit formulas for its copula and joint density are given. In the second part of the thesis, we study the problem of non-parametric estimation of maximum entropy densities of order statistics in Kullback-Leibler distance. Chapter 5 describes an aggregation method for probability and spectral densities, based on a convex combination of its logarithms, and shows non-asymptotic optimal bounds in deviation. In Chapter 6, we propose an adaptive method based on an exponential log-additive model to estimate the considered densities, and we show that it reaches the known minimax speeds. The application of this method to estimate the dimensions of defects is presented in Chapter 7.

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For the last two decades, most financial markets have undergone an evolution toward electronification. The market for corporate bonds is one of the last major financial markets to follow this unavoidable path. Traditionally quote-driven (i.e., dealer-driven) rather than order-driven, the market for corporate bonds is still mainly dominated by voice trading, but a lot of electronic platforms have emerged. These electronic platforms make it possible for buy-side agents to simultaneously request several dealers for quotes, or even directly trade with other buy-siders. The research presented in this article is based on a large proprietary database of requests for quotes (RFQ) sent, through the multi-dealer-to-client (MD2C) platform operated by Bloomberg Fixed Income Trading, to one of the major liquidity providers in European corporate bonds. Our goal is (i) to model the RFQ process on these platforms and the resulting competition between dealers, and (ii) to use our model in order to implicit from the RFQ database the behavior of both dealers and clients on MD2C platforms.

For the last two decades, most financial markets have undergone an evolution toward electronification. The market for corporate bonds is one of the last major financial markets to follow this unavoidable path. Traditionally quote-driven (i.e., dealer-driven) rather than order-driven, the market for corporate bonds is still mainly dominated by voice trading, but a lot of electronic platforms have emerged. These electronic platforms make it possible for buy-side agents to simultaneously request several dealers for quotes, or even directly trade with other buy-siders. The research presented in this article is based on a large proprietary database of requests for quotes (RFQ) sent, through the multi-dealer-to-client (MD2C) platform operated by Bloomberg Fixed Income Trading, to one of the major liquidity providers in European corporate bonds. Our goal is (i) to model the RFQ process on these platforms and the resulting competition between dealers, and (ii) to use our model in order to implicit from the RFQ database the behavior of both dealers and clients on MD2C platforms.

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This work focuses on the modeling of the dynamics of volatilities and correlations between financial assets returns. After a presentation of the literature on univariate and multivariate Garch models, the author establishes existence and uniqueness results for stationary solutions of dynamic correlation models of the DCC type (Engle, 2002). He then extends this class of models by including instantaneous volatilities and regime-switching probabilities in the dynamics of correlations. The new models are empirically evaluated on a portfolio of MSCI indices. Formal tests show that some of these new specifications improve the predictive power of the covariance matrix of returns and would be useful in portfolio management. Finally, focusing now on volatility risk, the author shows that hedging strategies of the main European equity indices based on implied volatility indices (VIX, VSTOXX) are relevant and allow both to hedge and reduce the equity risk of a portfolio.

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This thesis deals with density models for default times and their application to credit and counterparty risk. The first part is a theoretical contribution to the study of projections on different filtrations of the Radon-Nikodym density, in the form of Doleans-Dade exponential, occurring during measurement changes. The main result is the characterization of the measurement changes that preserve the immersion, obtained by applying our projection formulas. The second part aims at an informational dynamization of the static Gaussian copula model applied to a credit portfolio, which can be seen as a density model allowing to deal with CDO hedging by CDS or counterparty risk on credit derivatives. The main contributions are the introduction of the dynamic perspective, which gives a theoretical justification to the Gaussian copula bump-sensitivities used by practitioners, and the application to CVA calculations on a CDS.

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Generalizing the concepts introduced by dabrowska (1988), the author defines multidimensional hazard functions, associated with vectors of possibly independently right-censored durations. He proposes nonparametric estimators of these functions, by the convolution kernel method, and studies their statistical properties: almost sure, pointwise and uniform convergence on a pavement, convergence in law of the associated process. The main tool consists in an almost sure development of the estimator in a sum of independent and identically distributed variables plus a remainder. The speed of uniform convergence towards zero of this remainder is limited. Moreover, a choice of window, inspired by the one proposed by jones, marron and park (1991) in the case of density in dimension one, is studied in depth. The author shows that this choice is asymptotically optimal. Moreover, the speed of relative convergence of this window to the optimum in the sense of the integrated mean square deviation, takes place in root of n. Finally, the author shows that with positive kernels, it is not possible to obtain a higher speed in the previous case. If we work according to the criterion of the integrated mean square deviation, the corresponding relative speed is then n to the power of one tenth, the speed reached with the cross-validation method (Patil, 1993).