MAUME DESCHAMPS Veronique

Understanding the behaviour of environmental extreme events is crucial for evaluating economic losses, assessing risks, health care and many other aspects. In the spatial context, relevant for environmental events, the dependence structure plays a central rule, as it influence joined extreme events and extrapolation on them. So that, recognising or at least having preliminary informations on patterns of these dependence structures is a valuable knowledge for understanding extreme events. In this study, we address the question of automatic recognition of spatial Asymptotic Dependence (AD) versus Asymptotic independence (AI), using Convolutional Neural Network (CNN). We have designed an architecture of Convolutional Neural Network to be an efficient classifier of the dependence structure. Upper and lower tail dependence measures are used to train the CNN. We have tested our methodology on simulated and real data sets: air temperature data at two meter over Iraq land and Rainfall data in the east cost of Australia.

We propose a random forest based estimation procedure for Quantile Oriented Sensitivity Analysis-QOSA. In order to be efficient, a cross validation step on the leaf size of trees is required. Our full estimation procedure is tested on both simulated data and a real dataset.

We propose to study quantile oriented sensitivity indices (QOSA indices) and quantile oriented Shapley effects (QOSE). Some theoretical properties of QOSA indices will be given and several calculations of QOSA indices and QOSE will allow to better understand the behaviour and the interest of these indices.

We construct the Copula Recursive Tree (CORT) estimator: a flexible, consistent, piecewise linear estimator of a copula, leveraging the patchwork copula formalization and various piecewise constant density estimators. While the patchwork structure imposes a grid, the CORT estimator is data-driven and constructs the (possibly irregular) grid recursively from the data, minimizing a chosen distance on the copula space. The addition of the copula constraints makes usual denisty estimators unusable, whereas the CORT estimator is only concerned with dependence and guarantees the uniformity of margins. Refinements such as localized dimension reduction and bagging are developed, analyzed, and tested through applications on simulated data.

In this manuscript we propose a method for pricing insurance products that cover not only traditional risks, but also unforeseen ones. By considering the Poisson process parameter to be a mixed random variable, we capture the heterogeneity of foreseeable and unforeseeable risks. To illustrate, we estimate the weights for the two risk streams for a real dataset from a Portuguese insurer. To calculate the premium, we set the frequency and severity as distributions that belong to the linear exponential family. Under a Bayesian setup , we show that when working with a finite mixture of conjugate priors, the premium can be estimated by a mixture of posterior means, with updated parameters, depending on claim histories. We emphasise the riskiness of the unforeseeable trend, by choosing heavy-tailed distributions. After estimating distribution parameters involved using the Expectation-Maximization algorithm, we found that Bayesian premiums derived are more reactive to claim trends than traditional ones.

We propose a theoretical study of two realistic estimators of conditional distribution functions and conditional quantiles using random forests. The estimation process uses the bootstrap samples generated from the original dataset when constructing the forest. Bootstrap samples are reused to define the first estimator, while the second requires only the original sample, once the forest has been built. We prove that both proposed estimators of the conditional distribution functions are consistent uniformly a.s. To the best of our knowledge, it is the first proof of consistency including the bootstrap part. We also illustrate the estimation procedures on a numerical example.

Estimating high level quantiles of aggregated variables (mainly sums or weighted sums) is crucial in risk management for many application fields such as finance, insurance, environment. This question has been widely treated but new efficient methods are always welcome. especially if they apply in (relatively) high dimension. We propose an estimation procedure based on the checkerboard copula. It allows to get good estimations from a (quite) small sample of the multivariate law and a full knowledge of the marginal laws. This situation is realistic for many applications. Estimations may be improved by including in the checkerboard copula some additional information (on the law of a sub-vector or on extreme probabilities). Our approach is illustrated by numerical examples.

In the past decade, Sobol's variance decomposition have been used as a tool - among others - in risk management. We show some links between global sensitivity analysis and stochastic ordering theories. This gives an argument in favor of using Sobol's indices in uncertainty quantification, as one indicator among others.

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.

One of the main concerns in extreme value theory is to quantify the dependence between joint tails. Using stochastic processes that lack flexibility in the joint tail may lead to severe under-or over-estimation of probabilities associated to simultaneous extreme events. Following recent advances in the literature, a flexible model called max-mixture model has been introduced for modeling situations where the extremal dependence structure may vary with the distance. In this paper we propose a nonparametric model-free selection criterion for the mixing coefficient Our criterion is derived from a madogram, a notion classically used in geostatistics to capture spatial structures. The procedure is based on a nonlinear least squares between the theoretical madogram and the empirical one. We perform a simulation study and apply our criterion to daily precipitation over the East of Australia.

Natural disasters, such as heat waves, storms or extreme precipitation, originate from physical processes and have, by nature, a spatial or spatiotemporal dimension. The development of models and inference methods for these processes is a very active research area. This thesis deals with statistical inference for extreme events in the spatial and spatiotemporal framework. In particular, we are interested in two classes of stochastic processes: spatial max-mixing processes and spatio-temporal max-stable processes. We illustrate the results obtained on precipitation data in eastern Australia and in a region of Florida in the United States. In the spatial part, we propose two tests on the mixing parameter a of a spatial max-mixing process: the statistical test Za and the pairwise likelihood ratio LRa. We compare the performances of these tests on simulations. We use the pairwise likelihood for estimation. Overall, the performance of both tests is satisfactory. However, the tests encounter difficulties when the parameter a is on the boundary of the parameter space, i.e., a ∈ {0,1}, due to the presence of "nuisance" parameters that are not identified under the null hypothesis. We apply these tests in the context of an excess analysis over a large threshold for rainfall data in eastern Australia. We also propose a new estimation procedure to fit spatial max-mix processes when the extreme dependence class is not known. The novelty of this procedure is that it allows inference to be made without first specifying the family of distributions, thus letting the data speak for themselves and guide the estimation. In particular, the estimation procedure uses a least squares fit on the Fλ-madogram expression of a max-mix model that contains the parameters of interest. We show the convergence of the estimator of the mixture parameter a. An indication of asymptotic normality is given numerically. A simulation study shows that the proposed method improves the empirical coefficients for the class of max-mix models. We implement our estimation procedure on monthly rainfall maxima data in Australia for exploratory and confirmatory purposes. In the spatio-temporal part, we propose a semi-parametric estimation method for spatio-temporal max-stable processes based on an explicit expression of the spatio-temporal F-madogram. This part bridges the gap between geostatistics and extreme value theory. In particular, for regular grid observations, we estimate the spatio-temporal F-madogram by its empirical version and we apply a moment-based procedure to obtain the estimates of the parameters of interest. We illustrate the performance of this procedure by a study on simulations. Next, we apply this method to quantify the extremal behavior of maximum precipitation radar data in the state of Florida. This method can be an alternative or a first step for the composite likelihood. Indeed, the semi-parametric estimates could be used as a starting point for the optimization algorithms used in the pairwise likelihood method, in order to reduce the computation time but also to improve the efficiency of the method.

This project works with the risk model developed by Li et al. (2015) and quests modelling, estimating and pricing insurance for risks brought in by innovative technologies, or other emerging or latent risks. The model considers two different risk streams that arise together, however not clearly separated or observed. Specifically, we consider a risk surplus process where premia are adjusted according to past claim frequencies, like in a Bonus-Malus (BM) system, when we consider a classical or historical risk stream and an unforeseeable risk one. These are unknown risks which can be of high uncertainty that, when pricing insurance (ratemaking and experience rating), suggest a sensitive premium adjustment strategy. It is not clear for the actuary to observe which claim comes from one or the other stream. When modelling such risks it is crucial to estimate the behaviour of such claims, occurrence and their severity. Premium calculation must fairly reflect the nature of these two kinds of risk streams. We start proposing a model, separating claim counts and severities, then propose a premium calculation method, and finally a parameter estimation procedure. In the modelling we assume a Bayesian approach as used in credibility theory, a credibility approach for premium calculation and the use of the Expectation-Maximization (EM) algorithm in the estimation procedure.

Max-stable processes have been expanded to quantify extremal dependence in spatio-temporal data. Due to the interaction between space and time, spatio-temporal data are often complex to analyze. So, characterizing these dependencies is one of the crucial challenges in this field of statistics. This paper suggests a semiparametric inference methodology based on the spatio-temporal F-madogram for estimating the parameters of a space-time max-stable process using gridded data. The performance of the method is investigated through various simulation studies. Finally, we apply our inferential procedure to quantify the extremal behavior of radar rainfall data in a region in the State of Florida.

Max-mixture processes are defined as Z = max(aX, (1 − a)Y) with X an asymptotic dependent (AD) process, Y an asymptotic independent (AI) process and a ∈ [0, 1]. So that, the mixing coefficient a may reveal the strength of the AD part present in the max-mixture process. In this paper we focus on two tests based on censored pairwise likelihood estimates. We compare their performance through an extensive simulation study. Monte Carlo simulation plays a fundamental tool for asymptotic variance calculations. We apply our tests to daily precipitations from the East of Australia. Drawbacks and possible developments are discussed.

In [16], a new family of vector-valued risk measures called multivariate expectiles is introduced. In this paper, we focus on the asymptotic behavior of these measures in a multivariate regular variations context. For models with equivalent tails, we propose an estimator of these multivariate asymptotic expectiles, in the Fréchet attraction domain case, with asymptotic independence, or in the comonotonic case.

This paper proposes a stochastic approach to model temperature dynamic and study related risk measures. The dynamic of temperatures can be modelled by a mean-reverting process such as an Ornstein-Uhlenbeck one. In this study, we estimate the parameters of this process thanks to daily observed suprema of temperatures, which are the only data gathered by some weather stations. The expression of the cumulative distribution function of the supremum is obtained thanks to the law of the hitting time. The parameters are estimated by a least square method quantiles based on this function. Theoretical results, including mixing property and consistency of model parameters estimation, are provided. The parameters estimation is assessed on simulated data and performed on real ones. Numerical illustrations are given for both data. This estimation will allow us to estimate risk measures, such as the probability of heat wave and the mean duration of an heat wave.

No summary available.

The paper concerns quantile oriented sensitivity analysis. We rewrite the corresponding indices using the Conditional Tail Expectation risk measure. Then, we use this new expression to built estimators.

No summary available.

Kriging aims at predicting the conditional mean of a random field given the values of the field at some points. It seems natural to predict, in the same spirit as Kriging, other functionals. In our study, we focus on expectiles for elliptical random fields.

In this work, we consider an elliptical random field. We propose some spatial expectile predictions at one site given observations of the field at some other locations. To this aim, we first give exact expressions for conditional expectiles, and discuss problems that occur for computing these values. A first affine expectile regression predictor is detailed, an explicit iterative algorithm is obtained, and its distribution is given. Direct simple expressions are derived for some particular elliptical random fields. The performance of this expectile regression is shown to be very poor for extremal expectile levels, so that a second predictor is proposed. We prove that this new extremal prediction is asymptotically equivalent to the true conditional expectile. We also provide some numerical illustrations, and conclude that Expectile Regression may perform poorly when one leaves the Gaussian random field setting.

This paper is devoted to the introduction and study of a new family of multivariate elicitable risk measures. We call the obtained vector-valued measures multivariate expectiles. We present the different approaches used to construct our measures. We discuss the coherence properties of these multivariate expectiles. Furthermore, we propose a stochastic approximation tool of these risk measures.

This thesis presents contributions to multivariate modeling of distribution tails. We introduce a new modeling of the joint tail probabilities of a multivariate distribution with Pareto margins. This model is inspired by Wadsworth and Tawn (2013). A new non-standard regular multivariate variation of coefficient a bivariate function is introduced, allowing to generalize two modeling approaches respectively proposed by Ramos and Ledford (2009)and Wadsworth and Tawn (2013). Building on this modeling we propose a new class of semi-parametric models for multivariate extrapolation along paths spanning the entire first positive quadrant. We also consider parametric models built with a non-negative measure satisfying a constraint that generalizes that of Ramos and Ledford (2009). These new models are flexible and suitable for both dependence and asymptotic independence situations.

Probabilistic modeling of climate and environmental events must take into account their spatial nature. This thesis focuses on the study of risk measures for spatial processes. In a first part, we introduce risk measures able to take into account the dependency structure of the underlying spatial processes when dealing with environmental data. A second part is devoted to the estimation of parameters of max-mix processes. The first part of the thesis is dedicated to risk measures. We extend the work done in [44] on the one hand to Gaussian processes, on the other hand to other max-stable processes and to max-mix processes, other dependence structures are thus considered. The risk measures considered are based on the average L(A,D) of losses or damages D over a region of interest A. We then consider the expectation and variance of these normalized damages. First, we focus on the axiomatic properties of the risk measures, their computation and their asymptotic behavior (when the size of the region A tends to infinity). We compute the risk measures in different cases. For a Gaussian process, X, we consider the excess function: D+ X,u = (X-u)+ where u is a fixed threshold. For max-stable and max-mixed processes X, we consider the power function: DνX = Xν. In some cases, semi-explicit formulas for the corresponding risk measures are given. A simulation study tests the behavior of the risk measures with respect to the many parameters involved and the different forms of the correlation kernel. We also evaluate the computational performance of the different proposed methods. This one is satisfactory. Finally, we have used a previous study on pollution data in the Italian Piedmont, which can be considered as Gaussian. We study the risk measure associated to the legal pollution threshold given by the European directive 2008/50/EC. In a second part, we propose a procedure for estimating the parameters of a max-mix process, as an alternative to the composite maximum likelihood estimation method. This more classical method of estimation by composite maximum likelihood is especially efficient to estimate the parameters of the max-stable part of the mixture (and less efficient to estimate the parameters of the asymptotically independent part). We propose a least squares method based on the F-madogram: minimization of the quadratic difference between the theoretical F-madogram and the empirical F-madogram. This method is evaluated by simulation and compared to the composite maximum likelihood method. The simulations indicate that the F-madogram least squares method performs better in estimating the parameters of the asymptotically independent part.

Risk analysis has become a major issue in our society. Whatever the field of application in which a risky situation may occur, mathematics, and more particularly statistics and probabilities, are essential tools. The main purpose of this thesis is to develop relevant risk indicators and to study the extremal properties of processes involved in two fields of application: food risk and insurance. Risk theory is situated between extreme value analysis and the theory of random variables with regular variations or heavy tails. In the first chapter, we define the key elements of the risk theory as well as the notion of regular variation and we introduce different models related to food risk which will be studied in chapters 2 and 3. Chapter 2 presents the work done with Olivier Wintenberger. For classes of stochastic processes, under assumptions of regular variation, we develop a method that allows us to obtain asymptotic equivalents in a finite horizon of risk indicators in insurance and food risk such as the probability of ruin, the "time spent above a threshold" or the "severity of the ruin". Chapter 3 focuses on food risk models. Specifically, we study the extremal properties of different generalizations of a contaminant exposure process named KDEM for Kinetic Dietary Exposure Model proposed by Patrice Bertail and his co-authors in 2008. Under assumptions of regular variations, we propose asymptotic equivalents of the tail behavior and the extremal index of the exposure process. Finally, Chapter 4 reviews different statistical techniques particularly suited to study the extremal behavior of some Markov processes. Thanks to regenerative properties, it is possible to split the path of observations into independent and identically distributed blocks and thus study only the process on a block. These techniques apply even if the Markov chain is not atomic. We focus here on the estimation of the tail index and the extremal index. We illustrate the performance of these techniques by applying them on two models - in insurance and in finance - for which we know the theoretical results.

We proposed a semi-parametric estimation procedure in order to estimate the parameters of a max-mixture model and also of a max-stable model (inverse max-stable model) as an alternative to composite likelihood. A good estimation by the proposed estimator required the dependence measure to detect all dependence structures in the model, especially when dealing with the max-mixture model. We overcame this challenge by using the F-madogram. The semi-parametric estimation was then based on a quasi least square method, by minimizing the square difference between the theoretical F-madogram and an empirical one. We evaluated the performance of this estimator through a simulation study. It was shown that on an average, the estimation is performed well, although in some cases, it encountered some difficulties. We apply our estimation procedure to model the daily rainfalls over the East Australia.

In this paper, we consider isotropic and stationary max-stable, inverse max-stable and max-mixture processes $X=(X(s))_{s\in\bR^2}$ and the damage function $\cD_X^{\nu}= |X|^\nu$ with $0<\nu<1/2$. We study the quantitative behavior of a risk measure which is the variance of the average of $\cD_X^{\nu}$ over a region $\mathcal{A}\subset \bR^2$.} This kind of risk measure has already been introduced and studied for \vero{some} max-stable processes in \cite{koch2015spatial}. %\textcolor{red}{In this study, we generalised this risk measure to be applicable for several models: asymptotic dependence represented by max-stable, asymptotic independence represented by inverse max-stable and mixing between of them.} We evaluated the proposed risk measure by a simulation study.

In this work, we consider elliptical random fields. We propose some spatial quantile predictions at one site given observations at some other locations. To this aim, we first give exact expressions for conditional quantiles, and discuss problems that occur for computing these values. A first affine regression quantile predictor is detailed, an explicit formula is obtained, and its distribution is given. Direct simple expressions are derived for some particular elliptical random fields. The performance of this regression quantile is shown to be very poor for extremal quantile levels, so that a second predictor is proposed. We prove that this new extremal prediction is asymptotically equivalent to the true conditional quantile. Through numerical illustrations, the study shows that Quantile Regression may perform poorly when one leaves the usual Gaussian random field setting, justifying the use of proposed extremal quantile predictions.

Precipitation and streamflow are the two most important hydrometeorological variables for watershed analysis. They provide fundamental information for integrated water resources management, such as drinking water supply, hydropower, flood or drought forecasting, or irrigation systems.In this PhD thesis two distinct problems are addressed. The first one is based on the study of river flows. In order to characterize the global behavior of a watershed, long time series of flows covering several decades are necessary. However, missing data in the series represent a loss of information and reliability, and can lead to misinterpretation of the statistical characteristics of the data. The method we propose to address the flow imputation problem is based on dynamic regression models (DRM), more specifically, multiple linear regression coupled with ARIMA-type residual modeling. Contrary to previous studies involving the inclusion of multiple explanatory variables or the modeling of residuals from a simple linear regression, the use of DRMs allows both aspects to be taken into account. We apply this method to reconstruct daily streamflow data at eight stations located in the Durance watershed (France) over a 107-year period. By applying the proposed method, we succeed in reconstructing the flows without using other explanatory variables. We compare the results of our model with those obtained from a complex model based on analogues and hydrological modelling and from a nearest neighbor approach. In the majority of cases, DRMs show better performance when reconstructing missing data periods of different sizes, in some cases up to 20 years.The second problem we consider in this thesis concerns the statistical modeling of precipitation amounts. Research in this area is currently very active because the precipitation distribution exhibits a heavy upper tail and, at the beginning of this thesis, there was no satisfactory method to model the full range of precipitation. Recently, a new class of parametric distribution, called the extended generalized Pareto distribution (EGPD), has been developed for this purpose. This distribution exhibits better performance, but it lacks flexibility in modeling the central part of the distribution. In order to improve the flexibility, we develop two new models based on semiparametric methods. The first estimator developed first transforms the data with the cumulative EGPD distribution and then estimates the density of the transformed data by applying a nonparametric kernel estimator. We compare the results of the proposed method with those obtained by applying the parametric EGPD distribution on several simulations, as well as on two precipitation series in southeast France. The results show that the proposed method performs better than the EGPD, with the mean integrated absolute error (MIAE) of the density being in all cases almost two times lower.The second model considers a semiparametric EGPD distribution based on Bernstein polynomials. More precisely, we use a hollow mixture of beta densities. Similarly, we compare our results with those obtained by the parametric EGPD distribution on simulated and real data sets. As before, the MIAE of the density is significantly reduced, this effect being even more evident as the sample size increases.

This paper is devoted to the introduction and study of a new family of multivariate elicitable risk measures. We call the obtained vector-valued measures multivariate expectiles. We present the different approaches used to construct our measures. We discuss the coherence properties of these multivariate expectiles. Furthermore, we propose a stochastic approximation tool of these risk measures.

In finance, model risk is the risk of financial loss resulting from the use of models. It is a complex risk to apprehend and covers several very different situations, especially the estimation risk (a model generally uses an estimated parameter) and the model specification error risk (which consists in using an inadequate model). This thesis focuses on the quantification of model risk in the construction of rate or credit curves and on the study of the compatibility of Sobol indices with the theory of stochastic orders. It is divided into three chapters. Chapter 1 deals with the study of model risk in the construction of rate or credit curves. In particular, we analyze the uncertainty associated with the construction of rate or credit curves. In this context, we have obtained no-arbitrage bounds associated with implied default or rate curves that are perfectly compatible with the quotations of the associated reference products. In Chapter 2 of the thesis, we make the link between global sensitivity analysis and stochastic order theory. In particular, we analyze how the Sobol indexes transform following an increase in the uncertainty of a parameter in the sense of the stochastic dispersive order or excess wealth. Chapter 3 of the thesis focuses on the quantile contrast index. We first make the link between this index and the CTE risk measure, and then we analyze the extent to which an increase in the uncertainty of a parameter in the sense of stochastic dispersive order or excess wealth leads to an increase in the quantile contrast index. Finally, we propose a method for estimating this index. We show, under appropriate assumptions, that the estimator we propose is consistent and asymptotically normal.

In this work, we consider elliptical random fields. We propose some spatial quantile predictions at one site given observations at some other locations. To this aim, we first give exact expressions for conditional quantiles, and discuss problems that occur for computing these values. A first affine regression quantile predictor is detailed, an explicit formula is obtained, and its distribution is given. Direct simple expressions are derived for some particular elliptical random fields. The performance of this regression quantile is shown to be very poor for extremal quantile levels, so that a second predictor is proposed. We prove that this new extremal prediction is asymptotically equivalent to the true conditional quantile. Through numerical illustrations, the study shows that Quantile Regression may perform poorly when one leaves the usual Gaussian random field setting, justifying the use of proposed extremal quantile predictions.

The issue of capital allocation in a multivariate context arises from the presence of dependence between the various risky activities which may generate a diversification effect. Several allocation methods in the literature are based on a choice of a univariate risk measure and an allocation principle, others on optimizing a multivariate ruin probability or some multivariate risk indicators. In this paper, we focus on the latter technique. Using an axiomatic approach, we study its coherence properties. We give some explicit results in mono periodic cases. Finally we analyze the impact of the dependence structure on the optimal allocation.

European insurance sector will soon be faced with the application of the Solvency 2 regulation norms. It will create a real change in the risk management of insurance practices. The ORSA (Own Risk and Solvency Assessment) approach of the second pillar makes the capital allocation an important exercise for all insurers, especially when it comes to groups. Considering multi-branches firms, a capital allocation has to be based on multivariate risk modeling. Several allocation methods are present in the actuarial literature and insurance practices. In this paper, we focus on a risk allocation method. By minimizing some of the multivariate risk indicators, we study the coherence of the risk allocation using an axiomatic approach. Furthermore, we discuss what can be the best allocation choice for an insurance group.

In this paper we consider conditionally independent processes with respect to some dynamic factor. We derive some mixing properties for random processes when conditioning is given with respect to unbounded memory of the factor. Our work is motivated by some real examples related to risk theory.

The minimization of some multivariate risk indicators may be used as an allocation method, as proposed in Cénac et al. [6]. The aim of capital allocation is to choose a point in a simplex, according to a given criterion. In a previous paper [17] we proved that the proposed allocation technique satisfies a set of coherence axioms. In the present one, we study the properties and asymptotic behavior of the allocation for some distribution models. We analyze also the impact of the dependence structure on the allocation using some copulas.

The entry into force on January 1, 2016 of the European regulatory reform for the insurance sector, Solvency 2, is a historic event that will radically change risk management practices. It is based on an important consideration of the risk profile and vision, via the possibility of using internal models to calculate solvency capital and the ORSA (Own Risk and Solvency Assessment) approach for internal risk management. Mathematical modeling is thus an indispensable tool for a successful regulatory exercise. Risk theory must be able to accompany this development by proposing answers to practical problems, notably related to the modeling of dependencies and the choice of risk measures. In this context, this thesis presents a contribution to the improvement of actuarial risk management. In four chapters we present multivariate risk measures and their application to solvency capital allocation. The first part of this thesis is devoted to the introduction and study of a new family of elicitable multivariate risk measures that we will call multivariate expectations. The first chapter presents these measures and explains the different approaches used to construct them. The multivariate expectations verify a set of consistency properties that we also discuss in this chapter before proposing a tool for the stochastic approximation of these risk measures. The performances of this method being insufficient in the vicinity of the asymptotic levels of the thresholds of the expectiles, the theoretical analysis of the asymptotic behavior is necessary, and will be the subject of the second chapter of this part. The asymptotic analysis is carried out in a multivariate regular variation environment, it allows to obtain results in the case of equivalent marginal tails. We also present in the second chapter the asymptotic behavior of the multivariate expectiles under the previous assumptions in the presence of perfect dependence, or asymptotic independence, and we propose with the help of extreme value statistics estimators of the asymptotic expectile in these cases. The second part of the thesis focuses on the problem of solvency capital allocation in insurance. It is composed of two chapters in the form of published articles. The first one presents an axiomatization of the consistency of a capital allocation method in the most general framework possible, then studies the consistency properties of an allocation approach based on the minimization of multivariate risk indicators. The second paper is a probabilistic analysis of the behavior of the latter allocation approach as a function of the nature of the marginal risk distributions and the dependence structure. The asymptotic behavior of the allocation is also studied and the impact of the dependence is illustrated by different marginal models and different copulas. The presence of dependence between the different risks borne by insurance companies makes the multivariate approach a more appropriate answer to the different problems of risk management. This thesis is based on a multidimensional view of risk and proposes measures of a multivariate nature that can be applied to different actuarial problems of this nature.

In this paper, we study the quantitative behavior of a spatial risk measure corresponding to a damage function and a region, taking into account the spatial dependence of the underlying process. This kind of risk measure has already been introduced and studied for some max-stable processes in [Koch2015]. In this paper, we consider isotropic Gaussian processes and the excess damage function over a threshold. We performed a simulation study and a real data study.

In this paper we consider conditionally independent processes with respect to some dynamic factor. We derive some mixing properties for random processes when conditioning is given with respect to unbounded memory of the factor. Our work is motivated by some real examples related to risk theory.

Estimating high level quantiles of aggregated variables (mainly sums or weighted sums) is crucial in risk management for many application fields such as finance, insurance, environment. . . . This question has been widely treated but new efficient methods are always welcome. especially if they apply in relatively) high dimension. We propose an estimation procedure based on the checkerboard copula. It allows to get good estimations from a quite) small sample of the multivariate law and a full knowledge of the marginal laws. This situation is realistic for many applications, mainly in insurance. Moreover, we may also improve the estimations by including in the checkerboard copula some additional information (on the lawof a sub-vector or on extreme probabilities).

The approximation of a high level quantile or of the expectation over a high quantile (Value at Risk (VaR) or Tail Value at Risk (TVaR) in risk management) is crucial for the insurance industry. We propose a new method to estimate high level quantiles of sums of risks. It is based on the estimation of the ratio between the VaR (or TVaR) of the sum and the VaR (or TVaR) of the maximum of the risks. We use results on consistently varying functions. We compare the efficiency of our method with classical ones, on several models. Our method gives good results when approximating the VaR or TVaR in high levels on strongly dependent risks where at least one of the risks is heavy tailed.

The European insurance sector will soon be faced with the application of Solvency 2 regulation norms. It will create a real change in risk management practices. The ORSA approach of the second pillar makes the capital allocation an important exercise for all insurers and specially for groups. Considering multi-branches firms, capital allocation has to be based on a multivariate risk modeling. Several allocation methods are present in the literature and insurers practices. In this paper, we present a new risk allocation method, we study its coherence using an axiomatic approach, and we try to define what the best allocation choice for an insurance group is.

We prove a self normalized central limit theorem for a new mixing class of processes introduced in Kacem et al. (2013). This class is larger than the classical strongly mixing processes and thus our result is more general than Peligrad and Shao's (1995) and Shi's (2000) ones. The fact that some conditionally independent processes satisfy this kind of mixing properties motivated our study. We investigate the weak consistency as well as the asymptotic normality of the estimator of the variance that we propose.

In a multi-dimensional risk model with dependent lines of business, we propose to allocate capital with respect to the minimization of some risk indicators. These indicators are sums of expected penalties due to the insolvency of a branch while the global reserve is either positive or negative. Explicit formulas in the case of two branches are obtained for several models independent exponential, correlated Pareto). The asymptotic behavior (as the initial capital goes to infinity) is studied. For higher dimension and several periods, no explicit expression is available. Using a stochastic algorithm, we get estimations of the allocation, compare the different allocations and study the impact of dependence.

In this paper we derive explicit expressions for the probability of ruin in a renewal risk model with dependence described-by/incorporated-in the real-valued random variable Zk = −cτk + Xk , namely the loss between the (k − 1)–th and the k–th claim. Here c represents the constant premium rate, τk the inter-arrival time between the (k − 1)–th and the k–th claim and Xk is the size of the k–th claim. The dependence structure among (Zk )k>0 is given/driven by a Markov chain with a transition kernel satisfying an ordinary differential equation with constant coefficients.

This paper deals with the problem of estimating the tail of a bivariate distribution function. To this end we develop a general extension of the POT (Peaks-Over-Threshold) method, mainly based on a two-dimensional version of the Pickands-Balkema-de Haan Theorem. We introduce a new parameter that describes the nature of the tail dependence and we provide a way to estimate it. We construct a two-dimensional tail estimator and study its asymptotic properties. We also present real data examples which illustrate our theoretical results.

This thesis focuses on two different issues that have in common the contribution to modeling and risk management in actuarial science. In the first research theme, we are interested in the modeling of dependence in insurance and in particular, we propose an extension of the common factor models used in insurance. In the second research theme, we consider decreasing discrete distributions and study the effect of the addition of the convexity constraint on the convex extrema. Applications in connection with the theory of ruin motivate our interest in this subject. In the first part of the thesis, we consider a discrete time risk model in which the random variables are dependent but conditionally independent with respect to a common factor. In this dependence framework, we introduce a new concept for modeling the time dependence between risks in an insurance portfolio. Indeed, our modeling includes unbounded memory processes. More precisely, the conditioning is done with respect to a random vector of variable length over time. Under conditions of mixing the factor and a conditional mixing structure, we have obtained mixing properties for unconditional processes. With these results we can obtain interesting asymptotic properties. We note that in our asymptotic study it is rather the time that tends to infinity than the number of risks. We give asymptotic results for the aggregate process, which allows us to approximate the risk of an insurance company when time tends to infinity. The second part of the thesis deals with the effect of the convexity constraint on the convex extrema in the class of discrete distributions whose probability mass functions (p.m.f.) are decreasing on a finite support. The convex extrema in this class of distributions are well known. Our goal is to highlight how additional shape constraints of the convex type modify these extrema. Two cases are considered: the p.m.f. is globally convex on N and the p.m.f. is convex only from a given positive point. The corresponding convex extrema are computed using simple crossing properties between two distributions. Several illustrations in ruin theory are presented.

This paper deals with the problem of estimating the level sets of an unknown distribution function $F$. A plug-in approach is followed. That is, given a consistent estimator $F_n$ of $F$, we estimate the level sets of $F$ by the level sets of $F_n$. In our setting no compactness property is a priori required for the level sets to estimate. We state consistency results with respect to the Hausdorff distance and the volume of the symmetric difference. Our results are motivated by applications in multivariate risk theory. In this sense we also present simulated and real examples which illustrate our theoretical results.

Non-life actuarial science studies the various quantitative aspects of the insurance business. This thesis aims to explain from different perspectives the interactions between the different economic agents, the insured, the insurer and the market, in an insurance market. Chapter 1 highlights the importance of taking into account the market premium in the policyholder's decision to renew or not to renew his insurance contract with his current insurer. The need for a market model is established. Chapter 2 addresses this issue by using non-cooperative game theory to model competition. In the current literature, models of competition are always reduced to a simplistic optimization of premium volume based on a view of one insurer against the market. Starting from a one-period market model, a set of insurers is formulated, where the existence and uniqueness of the Nash equilibrium are verified. The properties of equilibrium premiums are studied to better understand the key factors of a dominant position of one insurer over the others. Then, the integration of the one-period game in a dynamic framework is done by repeating the game over several periods. A Monte-Carlo approach is used to evaluate the probability of an insurer being ruined, remaining leader, or disappearing from the game due to a lack of policyholders in its portfolio. This chapter aims at better understanding the presence of cycles in non-life insurance. Chapter 3 presents in depth the actual Nash equilibrium calculation for n players under constraints, called generalized Nash equilibrium. It provides an overview of optimization methods for solving the n optimization subproblems. This solution is done using a semi-smooth equation based on the Karush-Kuhn-Tucker reformulation of the generalized Nash equilibrium problem. These equations require the use of the generalized Jacobian for the locally Lipschitzian functions involved in the optimization problem. A convergence study and a comparison of the optimization methods are performed. Finally, chapter 4 deals with the calculation of the probability of ruin, another fundamental theme in non-life insurance. In this chapter, a risk model with dependence between the amounts or waiting times of claims is studied. New asymptotic formulas for the probability of ruin in infinite time are obtained in a broad framework of risk models with dependence between claims. In addition, explicit formulas for the probability of ruin in discrete time are obtained. In this discrete model, the dependence structure analysis allows to quantify the maximum deviation on the joint distribution functions of the amounts between the continuous and the discrete version.

The issue of surrender has long been of concern to insurers, particularly in the context of life insurance savings contracts, for which colossal sums are at stake. The emergence of the European Solvency II directive, which recommends the development of internal models (of which an entire module is dedicated to the management of surrender behavior risks), reinforces the need to deepen our knowledge and understanding of this risk. It is in this context that we address in this thesis the issues of segmentation and modeling of surrenders, with the objective of better understanding and taking into account all the key factors that influence policyholders' decisions. The heterogeneity of behaviors and their correlation, as well as the environment to which policyholders are subjected, are as many difficulties to be treated in a specific way in order to make forecasts. We have developed a methodology that has produced very encouraging results and has the advantage of being replicable by adapting it to the specificities of different product lines. Through this modeling, model selection appears as a central point. We address it by establishing the strong convergence properties of a new estimator, as well as the consistency of a new selection criterion in the context of mixtures of generalized linear models.

In a life insurance market in French-speaking sub-Saharan Africa that is lagging behind but has a bright future if endogenous technical and commercial solutions emerge, the thesis proposes theoretical and operational tools adapted to its development. This approach is in line with the actions undertaken by the regional supervisory authority (CIMA) to provide the region's insurers with appropriate tools. Indeed, CIMA has initiated work on the construction of new regulatory experience tables, which has provided reliable and relevant references for the mortality of the insured population in the region. However, some useful technical issues were not developed in this construction work. The thesis therefore gives them special attention. Thus, on the one hand, the thesis provides tools to account for differences in mortality between countries in the region, while limiting the systematic risks associated with sampling fluctuations (due to small data sample sizes per country). In particular, it appears that if independent modeling of each country is not appropriate, heterogeneity models with observable factors, such as the Cox or Lin and Ying model, can achieve this goal. However, it should be noted that these heterogeneity models do not eliminate the systematic risk of sampling fluctuations in the estimation of the model, but only reduce this risk while increasing the systematic risk of model selection. On the other hand, the thesis also provides tools to model future experience mortality in the region. In the absence of data on past trends in experience mortality, neither the classical Lee-Carter model nor its extensions are applicable. A solution based on a parametric adjustment, an assumption on the shape of the evolution of the mortality level (linear or exponential evolution) and an expert opinion on the generational life expectancy at a given age is then proposed (this work is based on the Bongaarts model). Then, in a second step, assuming the availability of data on past trends (which for the record is not the case at this stage in the region, but should be in the next few years), the thesis proposes a modeling of future mortality from an external mortality reference and an analysis of the associated systematic risks (risks related to sampling fluctuations and the choice of the mortality reference)

The purpose of this thesis is to develop some aspects of dependency modeling in risk management in dimension greater than one. The first chapter is a general introduction. The second chapter is an article entitled "Estimating Bivariate Tail: a copula based approach", submitted for publication. It concerns the construction of an estimator of the tail of a bivariate distribution. The construction of this estimator is based on a Peaks Over Threshold method and thus on a bivariate version of the Pickands-Balkema-de Haan Theorem. The modeling of the dependence is obtained via the Upper Tail Dependence Copula. We demonstrate convergence properties for the estimator thus constructed. The third chapter is based on a paper: "A multivariate extension of Value-at-Risk and Conditional-Tail-Expectation", submitted for publication. We address the problem of extending classical risk measures, such as Value-at-Risk and Conditional-Tail-Expectation, in a multidimensional setting using the multivariate Kendall function. Finally, in the fourth chapter of the thesis, we propose a contour estimator of a bivariate distribution function with a plug-in method. We demonstrate convergence properties for the estimators thus constructed. This chapter of the thesis is also constituted by an article, entitled "Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory", accepted for publication in the journal ESAIM:Probability and Statistics.

Initially, risk theory assumed independence between the different random variables and other parameters involved in actuarial modeling. Nowadays, this assumption of independence is often relaxed in order to take into account possible interactions between the different elements of the models. In this thesis, we propose to introduce dependence models for different aspects of risk theory. First, we suggest the use of copulas as a dependency structure. We first address a Tail-Value-at-Risk capital allocation problem for which we assume a copula-introduced link between different risks. We obtain explicit formulas for the capital to be allocated to the whole portfolio as well as the contribution of each risk when we use the Farlie-Gumbel-Morgenstern copula. For the other copulas, we provide an approximation method. In the second chapter, we consider the random process of the sum of the present values of the claims for which the random variables of the amount of a claim and the time elapsed since the previous claim are linked by a Farlie-Gumbel-Morgenstern copula. We show how to obtain explicit forms for the first two moments and then the mth moment of this process. The third chapter assumes another type of dependence caused by an external environment. In the context of the study of the probability of ruin of a reinsurance company, we use a Markovian environment to model the underwriting cycles. We first assume deterministic cycle phase change times and then consider them influenced in turn by the amounts of claims. Using the erlangization method, we obtain an approximation of the probability of ruin in finite time.

This thesis deals with the mixing properties of Markovian dynamical systems. The study of the associated transfer operator leads to estimates of the decay of correlations or mixing speed. These estimates allow to establish probabilistic theorems, for example the central limit theorem, for systems which do not possess, in general, the spectral hole property. The first part deals with Markov dynamics on a finite state space, associated with a non-Holderian potential. The decay of correlations depends on the continuity modulus of this potential. Moreover, these systems are stochastically stable. In a second part, we are interested in Markovian systems on a countable infinite state space. The decay of the correlations depends on the contribution to the transfer operator of the complementary of a finite number of cylinders. Effective estimates are given for non-uniformly dilating applications and for birth and death processes.