The development of flexible and interpretable statistical methods is necessary in order to provide appropriate risk assessment measures for extreme events and natural disasters. In this thesis, we address this challenge by contributing to the developing research field of Extreme-Value Theory. We initially study the performance of existing parametric and non-parametric estimators of extremal dependence for multivariate maxima. As the dimensionality increases, non-parametric estimators are more flexible than parametric methods but present some loss in efficiency that we quantify under various scenarios. We introduce a statistical tool which imposes the required shape constraints on non-parametric estimators in high dimensions, significantly improving their performance. Furthermore, by embedding the tree-based max-stable nested logistic distribution in the Bayesian framework, we develop a statistical algorithm that identifies the most likely tree structures representing the data's extremal dependence using the reversible jump Monte Carlo Markov Chain method. A mixture of these trees is then used for uncertainty assessment in prediction through Bayesian model averaging. The computational complexity of full likelihood inference is significantly decreased by deriving a recursive formula for the nested logistic model likelihood. The algorithm performance is verified through simulation experiments which also compare different likelihood procedures. Finally, we extend the nested logistic representation to the spatial framework in order to jointly model multivariate variables collected across a spatial region. This situation emerges often in environmental applications but is not often considered in the current literature. Simulation experiments show that the new class of multivariate max-stable processes is able to detect both the cross and inner spatial dependence of a number of extreme variables at a relatively low computational cost, thanks to its Bayesian hierarchical representation. These innovative methods and models are implemented to study the concentration maxima of various air pollutants and how these are related to extreme weather conditions for a number of sites in California, one of the most populated and polluted states of the US. As a result, we provide comprehensive measures of air quality that can be used by communities and policymakers worldwide to better assess and manage the health, environmental and financial impacts of air pollution extremes.

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Dans cette thèse, on considère une équation différentielle stochastique gouvernée par un processus de Lévy de saut pur dont l’indice d’activité des sauts α ∈ (0, 2) et on observe des données haute fréquence de ce processus sur un intervalle de temps fixé. Cette thèse est consacrée tout d’abord à l’étude du comportement de la densité du processus en temps petit. Ces résultats permettent ensuite de montrer la propriété LAMN (Local Asymptotic Mixed Normality) pour les paramètres de dérive et d’échelle. Enfin, on étudie des estimateurs de l’indice α du processus.La première partie traite du comportement asymptotique de la densité en temps petit du processus. Le processus est supposé dépendre d’un paramètre β = (θ,σ) et on étudie, dans cette partie, la sensibilité de la densité par rapport à ce paramètre. Cela étend les résultats de [17] qui étaient restreints à l’indice α ∈ (1,2) et ne considéraient que la sensibilité par rapport au paramètre de dérive. En utilisant le calcul de Malliavin, on obtient la représentation de la densité, de sa dérivée et de sa dérivée logarithmique comme une espérance et une espérance conditionnelle. Ces formules de représentation font apparaître des poids de Malliavin dont les expressions sont données explicitement, ce qui permet d’analyser le comportement asymptotique de la densité en temps petit, en utilisant la propriété d’autosimilarité du processus stable.La deuxième partie de cette thèse concerne la propriété LAMN (Local Asymptotic Mixed Normality) pour les paramètres. Le coefficient de dérive et le coefficient d’échelle dépendent tous les deux de paramètres inconnus et on étend les résultats de [17]. On identifie l’information de Fisher asymptotique ainsi que les vitesses optimales de convergence. Ces quantités dépendent de l’indice αLa troisième partie propose des estimateurs pour l’indice d’activité des sauts α ∈ (0,2) basés sur des méthodes de moments qui généralisent les résultats de Masuda [53]. On montre la consistence et la normalité asymptotique des estimateurs et on illustre les résultats par des simulations numériques / In this thesis, we consider a stochastic differential equation driven by a truncated pure jump Lévy process with index α ∈(0,2) and observe high frequency data of the process on a fixed observation time. We first study the behavior of the density of the process in small time. Next, we prove the Local Asymptotic Mixed Normality (LAMN) property for the drift and scaling parameters from high frequency observations. Finally, we propose some estimators of the index parameter of the process.The first part deals with the asymptotic behavior of the density in small time of the process. The process is assumed to depend on a parameter β = (θ,σ) and we study, in this part, the sensitivity of the density with respect to this parameter. This extends the results of [17] which were restricted to the index α ∈ (1,2) and considered only the sensitivity with respect to the drift coefficient. By using Malliavin calculus, we obtain the representation of the density, its derivative and its logarithm derivative as an expectation and a conditional expectation. These representation formulas involve some Malliavin weights whose expressions are given explicitly and this permits to analyze the asymptotic behavior in small time of the density, using the self-similarity property of the stable process.The second part of this thesis concerns the Local Asymptotic Mixed Normality property for the parameters. Both the drift coefficient and scale coefficient depend on the unknown parameters. Extending the results of [17], we compute the asymptotic Fisher information and find that the rate in the Local Asymptotic Mixed Normality property depends on the index α.The third part proposes some estimators of the jump activity index α ∈ (0,2) based on the method of moments as in Masuda [53]. We prove the consistency and asymptotic normality of the estimators and give some simulations to illustrate the finite-sample behaviors of the estimators

The statistical modeling of extreme natural hazards is becoming increasingly important due to climate change, whose effects have been increasingly visible throughout the last decades. It is thus crucial to understand the dependence structure of rare, high-impact events over space and time for realistic risk assessment. For spatial extremes, max-stable processes have played a central role in modeling block maxima. However, the spatial tail dependence strength is persistent across quantile levels in those models, which is often not realistic in practice. This lack of flexibility implies that max-stable processes cannot capture weakening dependence at increasingly extreme levels, resulting in a drastic overestimation of joint tail risk. To address this, we develop new dependence models in this thesis from the class of max-infinitely divisible (max-id) processes, which contain max-stable processes as a subclass and are flexible enough to capture different types of dependence structures. Furthermore, exact simulation algorithms for general max-id processes are typically not straightforward due to their complex formulations. Both simulation and inference can be computationally prohibitive in high dimensions. Fast and exact simulation algorithms to simulate max-id processes are provided, together with methods to implement our models in high dimensions based on the Vecchia approximation method. These proposed methodologies are illustrated through various environmental datasets, including air temperature data in South-Eastern Europe in an attempt to assess the effect of climate change on heatwave hazards, and sea surface temperature data for the entire Red Sea. In another application focused on assessing how the spatial extent of extreme precipitation has changed over time, we develop new time-varying $r$-Pareto processes, which are the counterparts of max-stable processes for high threshold exceedances.

Ce travail porte sur l'estimation des indices d'autosimilarité et de stabilité d'un processus ou champ stable fractionnaire et autosimilaire ou d'un processus stable multifractionnaire.Plus précisément, soit X un processus ou un champ stable H-autosimilaire à accroissements stationnaires (H-sssi) ou un processus stable multifractionnaire. Nous observons X aux points k/n, k=0,..., n.Nos estimations sont basées sur des variations de puissances négatives beta avec -1/2<beta<0: en effet, ces variations ont une espérance et une variance.Nous obtenons des estimateurs consistants, avec les vitesses de convergence, pour plusieurs processus H-sssi alpha-stables classiques (mouvement brownien fractionnaire, mouvement stable fractionnaire linéaire, processus de Takenaka, movement de Lévy).De plus, nous obtenons la normalité asymptotique de nos estimations pour le mouvement brownien fractionnaire et le mouvement de Lévy.Ce nouveau cadre nous permet de donner une estimation pour le paramètre d'autosimilarité H sans hypothèse sur alpha et, vice versa, nous pouvons estimer l'indice stable alpha sans hypothèse sur H.En généralisant, pour le cas d'une dimension supérieure à 1, nous obtenons également des estimateurs consistants pour H et alpha. Les résutats sont illustrés par des exemples: champ de Lévy fractionnaire, champ stable fractionnaire linéaire, champ de Takenaka.Pour les processus stables multifractionnaires, nous nous concentrons sur le mouvement brownien multifractionnaire et le processus stable multifractionnaire linéaire. Dans ces deux cas, nous obtenons la consistance des estimateurs pour la fonction d'autosimilarité à un temps donné u et pour l'indice stable alpha. / This work is concerned with the estimation of the self-similarity and the stability indices of a H-self-similarity stable process (field) or a multifractional stable process.More precisely, let X be a H-sssi (self-similar stationary increments) symmetric alpha-stable process (field) or a multifractional stable process. We observe X at points k/n, k=0,...,n.Our estimates are based on beta-negative power variations with -1/2<beta<0, thanks to the existence of expectations and covariances of these variations.We get consistent estimators, with rates of convergence, for several classical H-sssi alpha-stable processes(fractional Brownian motion, well-balanced linear fractional stable motion, Takenaka's processes, Lévy motion). Moreover, we get asymptotic normality of our estimates for fractional Brownian motion and Lévy motion.This new framework allows us to give an estimator for the self-similarity parameter H without assumptions on alpha and, vice versa, we can estimate the stable index alpha without assumptions on H.Generalizing for the case of high dimensions, we also obtain consistent estimators for H and alpha. The results are illustrated with some familiar examples: Lévy fractional Brownian field, well-balanced linear fractional stable field and Takenaka random field.For multifractional stable process, we concentrate on multifractional Brownian motion and linear multifractional stable process. In these two cases, we get the consistency of the estimators for the value of self-similarity function H at a fixed time u and for the stability index alpha.

Les extrêmes neigeux sont parmi les risques naturels les plus dangereux dans les régions montagneuses. Les processus max-stables, qui relient statistique des valeurs extrêmes et géostatistique, offrent un cadre approprié pour les étudier. Deux questions importantes concernant la dépendance spatiale des extrêmes sont traitées dans cette thèse à travers les cas des chutes et des hauteurs de neige dans les Alpes françaises : la sélection de modèle et la non-stationnarité temporelle. Nous utilisons pour cela deux jeux de données de maxima hivernaux de chutes de neige (90 stations de 1958 à 2013) et de hauteurs de neige (82 stations de 1970 à 2013). Nous décrivons d'abord une procédure de validation-croisée appropriée pour évaluer les capacités des processus max-stables à capturer la structure de dépendance des extrêmes spatiaux. Nous mettons en exergue trois processus max-stables pour leur aptitude à modéliser la dépendance spatiale des chutes de neige extrêmes : les processus de Brown-Resnick, géométrique gaussien et extrémal-t. Les performances de ces trois modèles sont extrêmement similaires, quel que soit le nombre de stations ou d'années. Ensuite, nous présentons une approche par fenêtre glissante pour évaluer l'évolution temporelle de la dépendance des extrêmes spatiaux. Nous montrons ainsi que les chutes de neige extrêmes ont tendance à être de moins en moins dépendantes spatialement. Nous montrons que cela est dû à une augmentation de la température provoquant une baisse du ratio neige/pluie. Il existe aussi un effet d'intensité avec des extrêmes moins dépendants à cause d'une baisse du cumul hivernal de chutes de neige. Enfin, nous présentons la première utilisation de processus max-stables avec des tendances temporelles dans la structure de dépendance spatiale. Cette approche est appliquée aux maxima de hauteurs de neige modélisés par un processus de Brown-Resnick. Nous montrons que leur dépendance spatiale est impactée par le changement climatique d'une manière similaire que celle des chutes de neige extrêmes. / Extreme snowfall and extreme snow depths are among the most dangerous hazards in the mountainous regions. Max-stable processes, which connect extreme value statistics and geostatistics by modeling the spatial dependence of extremes, offer a suitable framework to deal with. Two challenging issues concerning spatial dependence of extremes are broached in this thesis through the examples of snowfall and snow depths in the French Alps: model selection and temporal nonstationarity. We process two winter maxima data sets of 3-day snowfall (90 stations from 1958 to 2013) and snow depths (82 stations from 1970 to 2013). First, we introduce a leave-two-out cross-validation procedure appropriate for evaluating the predictive ability of max-stable processes to model the dependence structure of spatial extremes. We compare five of the most commonly used max-stable processes, using as a case study the snowfall maxima data set. This approach allows us to show that the extremal-t, geometric Gaussian and Brown-Resnick processes are able to represent as well the structure of dependence of the data, regardless of the number of stations or years. Then, we show, using a data-based approach allowing to make minimal modeling assumptions, that snowfall extremes tended to become less spatially dependent over time, with the dependence range reduced roughly by half during the study period. We demonstrate that this is attributable at first to the increase in temperature and its major control on the snow/rain partitioning. A magnitude effect, with less dependent extremes due to a decrease in winter cumulated snowfall, also exists. Finally, we tackle the first-ever use of max-stable processes with temporal trends in the spatial dependence structure. This approach is applied to snow depth winter maxima modeled by a Brown-Resnick process. We show that the spatial dependence of extreme snow depths is impacted by climate change in a similar way to that has been observed for extreme snowfall.

La prédiction de futurs évènements extrêmes est d’un grand intérêt dans de nombreux domaines tels que l’environnement ou la gestion des risques. Alors que la théorie des valeurs extrêmes univariées est bien connue, la complexité s’accroît lorsque l’on s’intéresse au comportement joint d’extrêmes de plusieurs variables. Un intérêt particulier est porté aux évènements de nature spatiale, définissant le cadre d’un nombre infini de dimensions. Sous l’hypothèse que ces évènements soient marginalement extrêmes, nous focalisons sur la structure de dépendance qui les lie. Dans un premier temps, nous faisons une revue des modèles paramétriques de dépendance dans le cadre multivarié et présentons différentes méthodes d’estimation. Les processus maxstables permettent l’extension au contexte spatial. Nous dérivons la loi en dimension finie du célèbre modèle de Brown- Resnick, permettant de faire de l’inférence par des méthodes de vraisemblance ou de vraisemblance composée. Nous utilisons ensuite des lois asymétriques afin de définir la représentation spectrale d’un modèle plus large : le modèle Extremal Skew-t, généralisant la plupart des modèles présents dans la littérature. Ce modèle a l’agréable propriété d’être asymétrique et non-stationnaire, deux notions présentées par les évènements environnementaux spatiaux. Ce dernier permet un large spectre de structures de dépendance. Les indicateurs de dépendance sont obtenus en utilisant la loi en dimension finie.Enfin, nous présentons une méthode d’estimation non-paramétrique par noyau pour les queues de distributions et l’appliquons à la sélection de modèles. Nous illustrons notre méthode à partir de l’exemple de modèles climatiques. / Projection of future extreme events is a major issue in a large number of areas including the environment and risk management. Although univariate extreme value theory is well understood, there is an increase in complexity when trying to understand the joint extreme behavior between two or more variables. Particular interest is given to events that are spatial by nature and which define the context of infinite dimensions. Under the assumption that events correspond marginally to univariate extremes, the main focus is then on the dependence structure that links them. First, we provide a review of parametric dependence models in the multivariate framework and illustrate different estimation strategies. The spatial extension of multivariate extremes is introduced through max-stable processes. We derive the finite-dimensional distribution of the widely used Brown-Resnick model which permits inference via full and composite likelihood methods. We then use Skew-symmetric distributions to develop a spectral representation of a wider max-stable model: the extremal Skew-t model from which most models available in the literature can be recovered. This model has the nice advantages of exhibiting skewness and nonstationarity, two properties often held by environmental spatial events. The latter enables a larger spectrum of dependence structures. Indicators of extremal dependence can be calculated using its finite-dimensional distribution. Finally, we introduce a kernel based non-parametric estimation procedure for univariate and multivariate tail density and apply it for model selection. Our method is illustrated by the example of selection of physical climate models.

This thesis focuses on the parameter stability of additive normal tempered stable processes when calibrating a volatility surface. The studied processes arise as a generalization of Lévy normal tempered stable processes, and their main characteristic are their time-dependent parameters. The theoretical background of the subject is presented, where its construction is discussed taking as a starting point the definition of Lévy processes. The implementation of an option valuation model using Fourier techniques and the calibration process of the model are described. The thesis analyzes the parameter stability of the model when it calibrates the volatility surface of a market index (EURO STOXX 50) during three time spans. The time spans consist of the periods from Dec 2016 to Dec 2017 (after the Brexit and the US presidential elections), from Nov 2019 to Nov 2020 (during the pandemic caused by COVID-19) and a more recent time period, April 2023. The findings contribute to the understanding of the model itself and the behavior of the parameters under particular economic conditions.

<p>This thesis, consisting of three papers and a summary, studies topics in the theory of stochastic processes related to long-range dependence. Much recent interest in such probabilistic models has its origin in measurements of Internet traffic data, where typical characteristics of long memory have been observed. As a macroscopic feature, long-range dependence can be mathematically studied using certain scaling limit theorems. </p><p>Using such limit results, two different scaling regimes for Internet traffic models have been identified earlier. In one of these regimes traffic at large scales can be approximated by long-range dependent Gaussian or stable processes, while in the other regime the rescaled traffic fluctuates according to stable ``memoryless'' processes with independent increments. In Paper I a similar limit result is proved for a third scaling scheme, emerging as an intermediate case of the other two. The limit process here turns out to be a non-Gaussian and non-stable process with long-range dependence.</p><p>In Paper II we derive a representation for the latter limit process as a stochastic integral of a deterministic function with respect to a certain compensated Poisson random measure. This representation enables us to study some further properties of the process. In particular, we prove that the process at small scales behaves like a Gaussian process with long-range dependence, while at large scales it is close to a stable process with independent increments. Hence, the process can be regarded as a link between these two processes of completely different nature.</p><p>In Paper III we construct a class of processes locally behaving as Gaussian and globally as stable processes and including the limit process obtained in Paper I. These processes can be chosen to be long-range dependent and are potentially suitable as models in applications with distinct local and global behaviour. They are defined using stochastic integrals with respect to the same compensated Poisson random measure as used in Paper II.</p>