LOISEL Stephane

In this paper, we extend the non-cooperative one-period game of Dutang et al. (2013) to model a non-life insurance market over several periods by considering the repeated (one-period) game. Using Markov chain methodology, we derive general properties of insurer portfolio sizes given a price vector. In the case of a regulated market (identical premium), we are able to obtain convergence measures of long run market shares. We also investigate the consequences of the deviation of one player from this regulated market. Finally, we provide some insights of long-term patterns of the repeated game as well as numerical illustrations of leadership and ruin probabilities.

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Predicting the evolution of mortality rates plays a central role for life insurance and pension funds. Standard single population models typically suffer from two major drawbacks: on the one hand, they use a large number of parameters compared to the sample size and, on the other hand, model choice is still often based on in-sample criterion, such as the Bayes information criterion (BIC), and therefore not on the ability to predict. In this paper, we develop a model based on a decomposition of the mortality surface into a polynomial basis. Then, we show how regularization techniques and cross-validation can be used to obtain a parsimonious and coherent predictive model for mortality forecasting. We analyze how COVID-19-type effects can affect predictions in our approach and in the classical one. In particular, death rates forecasts tend to be more robust compared to models with a cohort effect, and the regularized model outperforms the so-called P-spline model in terms of prediction and stability.

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In this paper, we propose and study a risk model with two types of claims in which the insurer may invest into a prevention plan which decreases the intensity of large claims without impacting the small claims. We identify a necessary and sufficient condition for insurers to use prevention if there is no surplus. If, in addition, the severity of large claims dominates that of small claims by the harmonic mean residual life (HMRL) order, insurers invest more in prevention in the presence of a surplus. Finally, we characterize the asymptotic optimal prevention strategy when the initial surplus tends to infinity in the two main cases where both claim types are light-tailed and where one of them is light-tailed and the other one is heavy-tailed.

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A high proportion of the kelp Laminaria hyperborea production is exported from kelp forests following seasonal storms or natural annual old blade loss. Transport of drifting kelp fragments can lead to temporary accumulations in benthic subtidal habitats. We investigated the degradation processes of L. hyperborea in a low subtidal sandy bottom ecosystem by setting up a 6‐month cage experiment to simulate accumulations of kelp fragments on the seafloor. We monitored temporal changes in biomass, nutritional quality (C:N ratio), respiration, quantum efficiency of photosystem II (Fv/Fm), bacterial colonization, and chemical defense concentrations. Biomass decomposition started after 2 weeks and followed a classic negative exponential pattern, leading to 50% degradation after 8 weeks. The degradation process seemed to reach a critical step after 11 weeks, with an increase in respiration rate and phlorotannin concentration in the tissues. These results likely reflect an increase in bacterial activity and a weakening of the kelp cell wall. After 25 weeks of degradation, only 16% of the initial biomass persisted, but the remaining large fragments looked intact. Furthermore, photosystems were still responding to light stimuli, indicating that photosynthesis persisted over time. Reproductive tissues appeared on some fragments after 20 weeks of degradation, showing a capacity to maintain the reproductive function. Our results indicate that L. hyperborea fragments degrade slowly. As they maintain major physiological functions (photosynthesis, reproduction, etc.) and accumulate on adjacent ecosystems, they may play a long‐term ecological role in coastal ecosystem dynamics.

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This paper is concerned with properties of Beta-unimodal distributions and their use to assess the basis risk inherent to index-based insurance or reinsurance contracts. To this extent, we first characterize s-convex stochastic orders for Beta-unimodal distributions in terms of the Weyl fractional integral. We then determine s-convex extrema for such distributions , focusing in particular on the cases s = 2, 3, 4. Next, we define an Enterprise Risk Management framework that relies on Beta-unimodality to assess these hedge imperfections , introducing several penalty functions and worst case scenarios. Some of the results obtained are illustrated numerically via a representative catastrophe model.

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This thesis deals with the implementation of prevention actions financed by an insurance company. It is composed of five chapters preceded by a general introduction which aims to present the difficulties linked to prevention, the tools used and the main results obtained. Chapter 1 proposes a method of unsupervised classification of health insurance policyholders into homogeneous risk groups, based on the benefits paid by an insurance company. This method has two phases: first, a dimension reduction of the data using positive matrix factorizations (PMF) is performed. The classification is then finalized using Kohonen maps. The tests of the method are also presented. The final classes obtained are finally analyzed in order to study whether some of them can be the object of a prevention action. A prevention action on psychiatry is proposed. Chapter 2 is a continuation of Chapter 1 since it focuses on comparing the quality of dimension reduction using NMF methods with that obtained using two other methods, the Word2Vec (W2V) and the marginalized stacked debugger autoencoders (mSDA). In particular, the stability of the final classifications is studied using a new stability measure. A complement on the consideration of temporality with the W2V algorithm is also presented. Chapter 3 proposes a study of the psychiatric risk within a complementary health organization, the algorithms of the previous chapters having allowed to identify this risk. With the help of a statistical study conducted on four databases, it is notably shown that insureds using psychiatry cost on average twice as much to the health insurer as an average individual. Some potential preventive actions are suggested in the conclusion. Chapter 4 focuses on the modelling of prevention within an insurance company. By integrating a prevention parameter into the compound Poisson model derived from the theory of ruin, it is indeed possible to measure the effect of prevention on certain indicators, such as the probability of ruin. Different optimal prevention strategies are proposed, and a sensitivity analysis is provided. Finally, chapter 5 proposes to extend the model considered in the previous chapter to the case where an insurance company is confronted with a light risk and a heavy risk. In such a model, the optimal prevention strategy depends on the amounts of reserves built up. Asymptotic results on the optimal strategies are provided.

Countries with common features in terms of social, economic and health systems generally have mortality trends which evolve in a similar manner. Drawing on this, many multi-population models are built on a coherence assumption which inhibits the divergence of mortality rates between two populations, or more, on the long run. However, this assumption may prove to be too strong in a general context, especially when it is imposed to a large collection of countries. We also note that the coherence hypothesis significantly reduces the spectrum of achievable mortality dispersion forecasts for a collection of populations when comparing to the historical observations. This may distort the longevity risk assessment of an insurer. In this paper, we propose a new model to forecast multiple populations assuming that the long-run coherent principle is verified by subgroups of countries that we call the "locally coherence" property. Thus, our specification is built on a trade-off between the Lee-Carter's diversification and Li-Lee's concentration features and allows to fit the model to a large number of populations simultaneously. A penalized vector autoregressive (VAR) model, based on the elastic-net regularization, is considered for modeling the dynamics of common trends between subgroups. Furthermore, we apply our methodology on 32 European populations mortality data and discuss the behavior of our model in terms of simulated mortality dispersion. Within the Solvency II directive, we quantify the impact on the longevity risk solvency capital requirement of an insurer for a simplified pension product. Finally, we extend our model by allowing populations to switch from one coherence group to another. We then analyze its incidence on longevity hedges basis risk assessment. JEL Classification: C18, C32, C53, J11.

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The estimation of R0 , the so-called "basic reproductive ratio", of the COVID-19 pandemic is of particular importance to help decision-makers take the necessary safeguard measures to protect the population. In this work, we examine a method based on the successive estimation of R0 over 3 non-overlapping periods (before lockdown, during lockdown and after). The approach is based on a variant of the, simple but flexible , SEIR compartmental model that allows to exploit the number of recovered individuals that are reported in the daily database published by national health agencies. The results of the approach is analysed w.r.t. data from France, at two levels of geographical subdivisions, i.e. the 13 regions and 96 departments that make up the metropolitan territory.

This paper is concerned with the quantification of basis risk in index-based insurance products using randomly scaled variables. To this extent, we first discuss the shape, the unimodality and the symmetry of randomly scaled variables depending on the distribution of the random scaling factor using Mellin transform. We explicitly obtain the distribution of a randomly scaled variable when the random scaling factor is either uniformly distributed or of Beta type. We then determine s-convex extremal distributions for randomly scaled variables and discuss the way of comparing it. Next, we define an Enterprise Risk Management framework that relies on randomly scaled variables to assess basis risk, introducing the class of generalized penalty functions. This ERM framework allows for setting up basis risk limits to eventually determine a Basis Risk Capital Requirement. The results are illustrated with particular cases that carefully challenge the methodology.

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In this paper, we propose and study a first risk model in which the insurer may invest into a prevention plan which decreases claim intensity. We determine the optimal prevention investment for different risk indicators. In particular, we show that the prevention amount minimizing the ruin probability maximizes the adjustment coefficient in the classical ruin model with prevention, as well as the expected dividends until ruin in the model with dividends. We also show that the optimal prevention strategy is different if one aims at maximizing the average surplus at a fixed time horizon. A sensitivity analysis is carried out. We also prove that our results can be extended to the case where prevention starts to work only after a minimum prevention level threshold.

The so-called market-consistency of the European life insurance valuation as shaped by regulation guidelines embeds numerous theoretical and practical misstatements.

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This doctoral dissertation focuses on the modeling of preventive effort and its relationship with market insurance. Each chapter attempts to capture different aspects of this problem, from the study of a criterion consistent with actuarial practices to the study of the supply side of insurance, including risk perception biases and an approach to prevention in dynamic time. Chapter 1 models the relationship between an insurer and an insured as a Stackelberg game. In this game, the insurer plays first by offering an insurance contract in the form of a loading factor. The insured then plays by choosing the optimal coverage rate and prevention effort. Both the insured and the insurer aim to minimize their respective risk measures, which are both consistent. The respective effects of self-insurance and self-protection on risk minimization will be studied. In each case, it will be shown that optimal choices for the insured exist and the optimal contract for the insurer will be characterized. Moreover, it will be shown that if the agent's risk measure decreases faster than his loss expectation, then the optimal effort is increasing with the loading factor with a potential discontinuity when the optimal coverage goes from full to zero. However, in the opposite case the optimal effort can be increasing or decreasing with the loading factor. Chapter 2 studies the relationship between self-insurance and market insurance also in the form of an optimization problem for one agent. Similar to Chapter 1, this agent must determine the coverage rate and the prevention effort that will optimally reduce its risk measure. The considered risk measure is called distortional and is defined from a non concave distortion function. This allows for potential individual cognitive biases in risk perception. The characterization of the optimal solution for the agent makes it possible to draw a new conclusion about the relationship between self-insurance and market insurance. Self-insurance is no longer just a substitute for market insurance, but can also be complementary to it, depending on the sensitivity of the prevention effort to the price of insurance. Chapter 3 focuses on self-protection by proposing a dynamic expected utility maximization problem. This takes the form of a stochastic control problem in which the agent chooses his insurance coverage and his prevention effort which is dynamic. The problem can be separated into two subproblems, the first one is an optimization in effort and the second one in insurance coverage. Since the individual wants to obtain the largest possible final wealth, he seeks to maximize the exponential utility expectation of this wealth. The agent's wealth can be seen as the solution of a backward-looking stochastic differential equation with a jump, this equation admits a unique solution and is moreover explicit. In particular, we obtain that the optimal self-protection effort is constant. The initial distribution of the loss process, when there is no effort, is given by a compound Poisson process which is in particular a Lévy process. Obtaining a constant optimal effort means that the Lévy property of the processes is preserved by maximizing an exponential utility expectation. The analysis of the problem in insurance coverage gives a sufficient condition to obtain the existence of an optimal level of coverage. The individual can then subscribe to an insurance policy by providing a preventive effort that will maximize his satisfaction or choose not to subscribe to the policy but by taking part in self-protection actions.

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The continuous increase in computer performance over the past decades has allowed for the widespread application of statistical learning theory in many fields. In particular, actuaries, historical experts in statistics, are increasingly turning to these innovative algorithms for the assessment of the risks they face. Thus, in this thesis, we examine how the integration of methodologies from statistical learning can contribute to the development of actuarial sciences and risk management through the study of three independent problems, presented in a general introduction. The first two chapters propose new mortality projection models in the context of the evaluation of longevity risk carried by insurance companies or pension funds. Chapter 1 focuses on the case where a single population is studied, while Chapter 2 extends the analysis to multi-populations. In both situations, the problem of high dimensionality appears central and we address it using a penalized vector autoregression (VAR). This model is applied directly on the mortality improvement rates in the first chapter, and on the time series resulting from the estimation of a Lee-Carter model for the second. The elastic-net penalty allows us to keep the great freedom of the space-time dependence structure offered by the VAR while remaining parsimonious in the number of parameters, and thus avoid overlearning. In Chapter 3 we analyze the surrender risk of life insurance contracts using supervised classification algorithms. Among others, we apply the wide margin separator (SVM) and the extreme gradient boosting (XGBoost). In order to compare the performances of the different classifiers, we adopt an economic vision from the marketing literature based on the potential profits of a retention campaign. We insist on the importance of the loss function retained in the statistical learning algorithms according to the objective sought: the use of a loss function in connection with the performance measure brings a significant improvement in the application of the XGBoost in our study. Finally, in the context of financial risk management, we study the dynamics of agricultural prices during particular trading sessions where government reports, containing valuable information for agents, are published. We examine the potential of open access data, in particular satellite images of vegetation index made available by NASA, for predicting market reactions. We then propose avenues of improvement to consider for practical implementation of this data enrichment methodology in risk management.

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In this paper, we propose and study a first risk model in which the insurer may invest into a prevention plan which decreases claim intensity. We determine the optimal prevention investment for different risk indicators. In particular, we show that the prevention amount minimizing the ruin probability maximizes the adjustment coefficient in the classical ruin model with prevention, as well as the expected dividends until ruin in the model with dividends. We also show that the optimal prevention strategy is different if one aims at maximizing the average surplus at a fixed time horizon. A sensitivity analysis is carried out. We also prove that our results can be extended to the case where prevention starts to work only after a minimum prevention level threshold.

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Sexual barriers associated with mate choice are often found to be associated with some level of ecological isolation between species. The independence and relative strength of sexual isolation are thus difficult to assess. Here we take advantage of a pair of marine isopod species (Jaera albifrons and J. praehirsuta) that show sexual isolation and coexist in populations where they share the same microhabitat or not (i.e. without or with ecological isolation). We estimated the strength of sexual isolation between J. albifrons and J. praehirsuta using no-choice trials and a multiple-choice experimental population. We found that sexual isolation is strong in both the presence and absence of ecological isolation, but that it is asymmetric and fails to prevent interspecific gene flow entirely. First-generation intrinsic post-zygotic barriers were low, and there was no sexual isolation within J. praehirsuta across habitats. The J. albifrons / J. praehirsuta species pair thus provides an example where the role of sexual isolation as a barrier to gene flow i) does not depend upon current ecological isolation, ii) seems to have evolved independently of local ecological conditions, but iii) is insufficient to complete speciation entirely on its own.

Modeling policyholders lapse behaviors is important to a life insurer since lapses affect pricing, reserving, profitability, liquidity, risk management, as well as the solvency of the insurer. Lapse risk is indeed the most significant life underwriting risk according to European Insurance and Occupational Pensions Authority's Quantitative Impact Study QIS5. In this paper, we introduce two advanced machine learning algorithms for lapse modeling. Then we evaluate the performance of different algorithms by means of classical statistical accuracy and profitability measure. Moreover, we adopt an innovative point of view on the lapse prediction problem that comes from churn management. We transform the classification problem into a regression question and then perform optimization, which is new for lapse risk management. We apply different algorithms to a large real-world insurance dataset. Our results show that XGBoost and SVM outperform CART and logistic regression, especially in terms of the economic validation metric. The optimization after transformation brings out significant and consistent increases in economic gains.

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In the summer of 2013, the term "Big Data" made its appearance and aroused strong interest among companies. This thesis studies the contribution of these methods to actuarial sciences. It addresses both theoretical and practical issues on high-potential topics such as textit{Optical Character Recognition} (OCR), text analysis, data anonymization or model interpretability. Starting with the application of machine learning methods in the calculation of economic capital, we then try to better illustrate the frontality that can exist between machine learning and statistics. Putting forward some advantages and different techniques, we then study the application of deep neural networks in the optical analysis of documents and text, once extracted. The use of complex methods and the implementation of the General Data Protection Regulation (GDPR) in 2018 led us to study the potential impacts on pricing models. By applying anonymization methods on pure premium calculation models in non-life insurance, we explored different generalization approaches based on unsupervised learning. Finally, as the regulation also imposes criteria in terms of model explanation, we conclude with a general study of the methods that allow today to better understand complex methods such as neural networks.

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In this paper, we propose and study a risk model with two types of claims in which the insurer may invest into a prevention plan which decreases the large claims intensity without impacting the small claims. In this setting, we prove that prevention is advantageous when claim severities for small and large claims are ordered in the sense of the Harmonic-Mean-Residual-Lifetime (HMRL) order. In addition, we show that the optimal prevention amount is the lowest when there is no initial surplus. Finally, we characterize the asymptotic optimal prevention strategy when the initial surplus tends to infinity in the two main cases where both claim types are light-tailed and where one of them is light-tailed and the other one is heavy-tailed.

On paper, prevention appears to be a good complement to health insurance. However, its implementation is often costly. To maximize the impact and efficiency of prevention plans these should target particular groups of policyholders. In this article, we propose a way of clustering policyholders that could be a starting point for the targeting of prevention plans. This two-step method mainly classifies using policyholder health consumption. This dimension is first reduced using a Nonnegative matrix factorization algorithm, producing intermediate health-product clusters. We then cluster using Kohonen's map algorithm. This leads to a natural visualization of the results, allowing the simple comparison of results from different databases. We apply our method to two real health-insurer datasets. We carry out a number of tests (including tests on a text-mining database) of method stability and clustering ability. The method is shown to be stable, easily-understandable, and able to cluster most policyholders efficiently.

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In the insurance sector, the latest regulatory developments and accounting standards are moving towards standardization of risk management within organizations. In this context, the main objective of my thesis is to propose different methodologies for risk assessment and analysis in this sector. The first part of this manuscript deals with the problem of individual provisioning in non-life. I propose adaptations of ensemblistic machine learning algorithms and of some performance metrics for estimating claim durations and ultimate claim expenses in the presence of right-censored data. The application of these methods to real data from loan contracts or group insurance contracts leads to better and more robust estimates of the considered parameters. The second part presents an approach for estimating a one-year shock on the Undertaking Specific Parameters of the life insurance module of pillar 1 of the Solvency II standard formula. The use of the American credibility (or limited variation credibility) allows the partial taking into account of the constraints of availability of experience data (volumetry and depth of history) when calibrating the shocks. As an illustration, I have applied this approach to the incidence and continuation (or reinstatement) risks of disability and out-of-work benefits in a portfolio of loan contracts. The results obtained show significant decreases in the solvency capital requirements (SCR) of the underwriting risk compared to the standard formula. The third part is a descriptive study of the calculations of the standard formula for the evaluation of the economic capital requirement of the dependency risk. It highlights the shortcomings of the standard and suggests ways to improve it in order to better take into account the specificities of this risk. Finally, in the last part of the manuscript, I propose a comparative study of the preferences of attitudes towards risk in the financial sector, in particular banking and insurance. This is an empirical analysis conducted in three geographical areas (America, Europe and Africa) in order to measure the links and differences between risk attitude profiles and certain socio-demographic variables.

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The objective of this thesis is to analyze the interactions between economic agents operating in the retail insurance market. On the one hand, policyholders wishing to cover themselves against a risk of loss must explore the market in order to subscribe to a contract in line with their perception of the risk. On the other hand, insurers compete in a regulated market, imposing on them a certain level of capital in order to guarantee their solvency in a context of uncertainty about the risks underwritten. On the other hand, intermediaries offer their services in order to facilitate the interaction between consumers, who are averse to risk, and firms, which take risks. It is therefore in this context that we analyze the behavior of insurance actors from different perspectives. Chapters 1 and 2 of this thesis are the result of laboratory experiments, carried out using a web interface designed specifically for these studies. The results in Chapter 3 are based on a theoretical model and numerical simulations. Chapter 1 focuses on the relationship between honesty and honesty beliefs of economic agents. Using data collected in the laboratory, we show how uncertainty and the feeling of being in more or less advantageous conditions impact both the level of honesty and the belief in honesty towards others. In general, consumers overestimate the honesty of intermediaries. Thus, this result justifies their presence in the insurance market. On the other hand, we also show that the financial incentives offered to intermediaries distort honesty beliefs. The lower the incentive level, the more dishonest behavior is anticipated by consumers. In Chapter 2, we highlight the dilemma faced by the consumer in a market with multiple distribution channels. Should he explore by himself and choose among a large set of contracts or delegate part of his decision to an intermediary plus or minus search costs, we show that obfuscation related to a large amount of information and beliefs in the honesty of intermediaries are the main determinants of search and purchase decisions. We also show that obfuscation and intermediaries' attitudes are sources of inefficiency in decision making, in particular with respect to the characteristics of the insurance contracts purchased by consumers. In this sense, the identification of a focus effect supports the importance of the price level in decision making to the detriment of the risk environment and the level of coverage. The introduction of search costs in the exploration process, as well as the heterogeneity of honesty beliefs, justify the multi-channel distribution strategies adopted by insurers. An analysis of a repeated non-cooperative game is presented in Chapter 3 of this thesis where losses and consumer behavior are stochastic and insurers compete on price. In order to incorporate the regulators' constraints, we determine Nash equilibria under solvency constraints. We also analyze the sensitivity of equilibrium premiums to the parameters of the game, in particular when firms do not benefit from the same comparative advantages (i.e. reputation leading to different levels of customer retention, insurers' seniority leading to different capital stocks).

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The human lifespan has been increasing in the world for the last few centuries. This increase has been greater than predicted by specialists who have set limits. Despite significant uncertainties about the future of longevity, the biology of aging and its applications seem to be on the way to lowering mortality rates in old age, similar to the drop in infant mortality rates 150 years ago. The pharmaceutical industry is becoming aware of the potential of biomedical innovations stemming from the biology of aging, buying up biotechs and developing in-house teams. However, the actuarial tables, like the Lee Carter model, tend to predict an artificial deceleration of longevity and the calculated risks are far from representing major advances in the biology of aging. Future mortality models are developed here without producing this deceleration. It appears that an increase of about one quarter per year has so far been a better predictor than the trends in each country. Other models predict accelerations. We estimate the impacts on pensions. Ongoing pharmaceutical efforts to apply the results of biomedical research can be feared because of their impacts on pensions. We study the extent to which a longevity mega-fund can both help finance funded pensions and a large number of pharmaceutical developments: the pooling of clinical risks can financially capture longevity-related biomedical successes.

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As more and more people believe that significant life extensions may come soon, should commonly used future mortality assumptions be considered prudent? We find here that commonly used actuarial tables for annuitants – as well as the Lee-Carter model – do not extrapolate life expectancy at the same rate for future years as for past years. instead they produce some longevity deceleration. This is typically because their mortality improvements decrease after a certain age, and those age-specific improvements are constant over time. As potential alternatives i) we study the Bongaarts model that produces straight increases in life expectancy. ii) we adapt it to produce best-practice longevity trends iii) we compare with various longevity scenarios even including a model for “life extension velocity”. iv) after gathering advances in biogerontology we discuss elements to help retirement systems resist to a potential strong increase in life expectancy.

The ORSA Own Risk Solvency and Assessment is a set of rules defined by the European Solvency II directive. It is intended to serve as a decision support tool and strategic risk analysis. In the context of the ORSA, insurance companies must assess their future solvency, on an ongoing and prospective basis. In order to do so, they must obtain projections of their balance sheet (assets and liabilities) over a certain time horizon. In this thesis, we focus mainly on the aspect of predicting future asset values. More precisely, we deal with the yield curve, its construction and extrapolation at a given date, and its predictions envisaged in the future. In the text, we refer to the "yield curve", but it is in fact the construction of discount factor curves. Counterparty default risk is not explicitly addressed, but techniques similar to those developed can be adapted to the construction of rate curves incorporating counterparty default risk.

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This thesis is composed of three chapters that deal with impulse control problems. In the first chapter, we introduce a general framework for impulse control with uncertainty. Knowing an a priori law on unknown parameters, we explain how it should evolve and integrate it to the optimal control problem. We characterize the solution through a quasivariational parabolic equation that can be solved numerically and give examples of applications to finance. In the second chapter, we introduce an impulse control problem with uncertainty in an actuarial setting. An (re)insurer faces natural catastrophes and can issue CAT bonds to reduce the risk taken. We again characterize the optimal control problem through a numerically solvable quasi-variational parabolic equation and give some application examples. In the last chapter, we propose a model of the price through a completely endogenous order book. We solve impulse optimal control problems (order placement) of rational economic agents that we gather on a same market.

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This paper aims to shed light on the dilemma faced by insurance purchasers faced with multiple distribution channels. Should the consumer herself choose from a large set of insurance policies or rather delegate a part her decision to an intermediary who is more or less honest? We consider decisions based on a number of real-world insurance distribution channels with different information frames. Beliefs about intermediary honesty are the main determinants of individual choice. In addition, the obfuscation of information is the main source of inefficiency in decision-making, particularly regarding the characteristics of the insurance contracts chosen by consumers.

In this paper, our aim is to measure mortality rates which are specific to individual observable factors when these can change during life. The study is based on longitudinal data recording marital status and socio-professional features at census times, therefore the observation scheme is interval-censored since individual characteristics are only observed at isolated dates and transition times remain unknown. To this aim, we develop a parametric maximum likelihood estimation procedure for multi-state models that takes into account both interval-censoring and reversible transitions. This method, inspired by recent advances in the statistical literature, allows us to capture characteristic-specific mortality rates, in particular to recover the mortality compensation law at high ages, but also to capture the age pattern of characteristics changes. The dynamics of several population compositions is addressed, and allows us to give explanations on the pattern of aggregate mortality, as well as on the impact on typical life insurance products. Particular attention is devoted to characteristics changes and parameter uncertainty that are both crucial to take into account.

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The Solvency II directive has introduced a specific so-called risk-neutral framework to valuate economic accounting quantities throughout European life insurance companies. The adaptation of this theoretical notion for regulatory purposes requires the addition of a specific criterion, namely the market-consistency, in order to objectify the choice of the valuation probability measure. This paper aims at pointing out and fixing some of the major risk sources embedded in the current regulatory life insurance valuation scheme. We compare actuarial and financial valuation schemes. We then address first operational issues and potential market manipulation sources in life insurance, induced by both theoretical and regulatory pitfalls. For example, we show that calibrating the interest rate model in October 2014 instead of December 31 st 2014 generates a 140%-increase in the economic own funds of a representative French life insurance company. We propose various adaptations of the current implementations, including product-specific valuation scheme, to limit the impact of these market-inconsistencies.

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The Solvency II directive has introduced a specific so-called risk-neutral framework to valuate economic accounting quantities throughout European life insurance companies. The adaptation of this theoretical notion for regulatory purposes requires the addition of a specific criterion, namely the market-consistency, in order to objectify the choice of the valuation probability measure. This paper aims at pointing out and fixing some of the major risk sources embedded in the current regulatory life insurance valuation scheme. We compare actuarial and financial valuation schemes. We then address first operational issues and potential market manipulation sources in life insurance, induced by both theoretical and regulatory pitfalls. For example, we show that calibrating the interest rate model in October 2014 instead of December 31 st 2014 generates a 140%-increase in the economic own funds of a representative French life insurance company. We propose various adaptations of the current implementations, including product-specific valuation scheme, to limit the impact of these market-inconsistencies.

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Most mortality models are generally calibrated on national population. However, pensions funds and annuity providers are mainly interested in the mortality rates of their own portfolio. In this paper we put forward a multivariate approach for forecasting pairwise mortality rates of related population. The investigated approach links national population mortality to a subset population using an econometric model that captures a long-run relationship between both mortality dynamics. This model does not lay the emphasis on the correlation that the two given mortality dynamics would present but rather on the long-term behaviour, which suggests that the two time-series cannot wander off in opposite directions for very long without mean reverting force on grounds of biological reasonableness. The model additionally captures the short-run adjustment between the considered mortality dynamics. Our aim is to propose a consistent approach to forecast pairwise mortality and to some extent to better control and assess basis risk underlying index-based longevity securitization. An empirical comparison of the forecast of one-year death probabilities of portfolio-experienced mortality is performed using both a factor-based model and the proposed approach. The robustness of the model is tested on mortality rate data for England & Wales and Continuous Mortality Investigation assured lives representing a sub-population.

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This paper deals with finite sequences of exchangeable 0–1 random variables. Our main purpose is to exhibit the dependence structure between such indicators. Working with Kendall's representation by mixture, we prove that a convex order of higher degree on the mixing variable implies a supermodular order of same degree on the indicators, and conversely. The convex order condition is then discussed for three standard distributions (binomial, hypergeometric and Stirling) in which the parameter is randomized. Distributional properties of exchangeable indicators are also revisited using an underlying Schur-constant property. Finally, two applications in insurance and credit risk illustrate some of the results.SCOPUS: ar.jinfo:eu-repo/semantics/publishe.

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The purpose of this thesis is to link the valuation of derivatives to the scenario analysis that is the reference in bank management. The valuation of options is an area that has given rise to a very large literature with a well-defined mathematical framework, developed rather for market finance practitioners with a view to replication hedging. The risk management of these products is therefore analyzed from the perspective of the famous principle of no arbitrage opportunity. According to this principle, the estimate for a future rate becomes the one anticipated by the markets today thanks to the presence of hedging instruments. On the other hand, corporate financial management is based on a so-called "scenario approach", in which different scenarios that seem insightful and make economic sense to the bank's management are selected, and a projection for each scenario is studied. These scenarios are generally not probabilized. We are in a model of the uncertain where the notion of probability and expectation does not intervene. The bank, which has complex products related to its activity, does not hedge its risks in the same way as a service related to market activities. The study of this framework of analysis is part of the thesis. We analyze this need for projection by scenario on the perimeter of complex products.

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Technical risk management in French-speaking sub-Saharan Africa is a notion that is often absent in practice. Indeed, in this zone, it is very easy to find banks, insurance companies and pension institutions conducting their activities without integrating risk1 into their core management. This situation explains, a priori, the absence of reliable databases for quantitative studies. This thesis, which is complementary to the work of Kamega A2., focuses on the design of relevant actuarial tools adapted to the technical management of risks in French-speaking sub-Saharan Africa, which can be used by the governments of this zone as well as in the insurance and banking industries. In view of the progressive development of the countries in the CIPRES zone, we believe that the economic scenario generator (ESG) is the common tool for the technical management of risks related to the activities of governments and the banking and insurance industry. Note that the ESG is a tool capable of projecting economic and financial variables into a coherent system. This rich information will allow, for example, the governments of these countries to elaborate their budgets, to mobilize resources on the local financial market and to technically manage public debt. In the context of the design of the GSE, the contribution of this thesis consists in first specifying mathematical models, adapted to the context of the CIPRES zone, covering a large number of economic and financial variables. In a second step, calibration methods are presented in the context of absence of data (expert opinion) or presence of data (statistical approaches). Particular attention is paid to the extension of the GSE in order to take into account the future needs of professionals in the CIPRES area. This thesis also gives importance to the application of the GSE in the development of CIPRES countries through the contribution of the yield curve in the analysis and the conduct of monetary policy, the forecasting of economic and financial quantities, the estimation of implicit default probabilities and recovery rates of States and companies in a context of rating in local currency and the application of the Basel II/III framework in 2018. In the context of pension plans, these actuarial tools are useful in determining the parameters for steering the plan, in particular the "best estimate" valuation of the plan's liabilities, the financing and the asset allocation strategy.

This thesis deals with the modeling of population dynamics and its applications to demography and actuarial science on the one hand, and to the study of Hawkes processes on the other hand. This thesis proposes to explore through different points of view how the structure of a population is deformed, both in terms of age distribution and in terms of its composition in terms of characteristics. Through five chapters, we present the same philosophy which, in order to understand how aggregate quantities evolve, proposes to study the dynamics of the population at a finer scale, that of the individual. After a first introductory chapter in French, detailing the motivations and the main contributions, we first propose in Chapter 2 the description of the general framework of the random dynamic modeling of populations structured in characteristics and ages, based on Bensusan et al. (2010-2015), as well as several examples motivated by demographic and actuarial applications. We detail the mathematical construction of such processes as well as the link to classical deterministic equations in demography. We also discuss the impact of heterogeneity on the example of a cohort effect, as well as the role of the random environment. The next two chapters highlight the importance of the age pyramid. The general population model from Chapter 2 is declined in Chapter 3 to study Hawkes processes with general immigrants, for which we exploit the concept of the age pyramid. In this theoretical study, based on Boumezoued (2015b), we establish new results on their distribution for a class of functions that generalize the exponential case studied so far. In Chapter 4, following Arnold et al. (2015), we analyze the impact of changes in cause-of-death mortality on the dynamics of the population pyramid, and in particular on the dependency ratio which is a crucial indicator of population aging. By including the set of births in the dynamics, this simulation work, based on WHO data, complements the existing literature on causes of death which traditionally focuses on mortality indicators. The last two chapters focus on population heterogeneity. Chapter 5, based on Boumezoued et al. (2015), proposes to measure mortality heterogeneity in INSEE Permanent Demographic Sample data. As part of this contribution of adapting statistical methods and its implementation on real data, we propose a parametric maximum likelihood estimation method for multi-state models that takes into account both interval censoring, characteristic of longitudinal data from the census, and also the return in intermediate states. Finally, Chapter 6, taken from Boumezoued (2015a), repeats the general model from Chapter 2 in which individuals can give birth, change characteristics, and die. The contribution of this theoretical part is to study the behavior of the population when individual characteristics change frequently. We establish a large population limit theorem for the population pyramid process, whose behavior is then described by birth and death rates aggregated over the stable structure in terms of characteristics.

The aim of this experiment is to provide tools that allow characterisation of waste biodegradation state at a laboratory scale. Four geophysical methods, self-potential, electrical resistivity, temporal and spectral induced polarization, will be monitored from the beginning to the end of municipal solid waste biodegradation cycle. For this reason five columns were filled with waste and equipped with measurement electrodes. The measurements were started in February and will continue until the waste has completely degraded.

In this paper, we introduce a new structured financial product: the so-called Life Nominal Chooser Swaption (LNCS). Thanks to such a contract, insurers could keep pure longevity risk and transfer a great part of interest rate risk underlying annuity portfolios to financial markets. Before the issuance of the contract, the insurer determines a confidence band of survival curves for her portfolio. An interest rate hedge is set up, based on swaption mechanisms. The bank uses this band as well as an interest rate model to price the product. At the end of the first period (e.g. 8 to 10 years), the insurer has the right to enter into an interest rate swap with the bank, where the nominal is adjusted to her (re-forecasted) needs. She chooses (inside the band) the survival curve that better fits her anticipation of future mortality of her portfolio (during 15 to 20 more years, say) given the information available at that time. We use a population dynamics longevity model and a classical two-factor interest rate model %two-factor Heath-Jarrow-Morton (HJM) model for interest rates to price this product. Numerical results show that the option offered to the insurer (in terms of choice of nominal) is not too expensive in many real-world cases. We also discuss the pros and the cons of the product and of our methodology. This structure enables insurers and financial institutions to remain in their initial field of expertise.

The Solvency II Directive, submitted by the European Commission in 2009, came into force in January 2016. It is based on three pillars. The first pillar deals with quantitative requirements related to the calculation of the Solvency Capital Requirement. The second pillar deals with risk governance. The third pillar deals with required documents and information, market discipline. For life insurance, the quantitative requirements (Pillar I and part of Pillar II) introduce a high level of complexity. Indeed, in order to create a system adapted to the specificities of companies, the directive has introduced a framework for the valuation of the balance sheet of insurers that is very delicate to understand and use, the economic valuation. Because of this complexity, most European life insurers have, during their first years of implementing the directive, chosen to focus on pillar I knowing that the calculation of the capital requirement would be an essential part of the system. In this thesis, I have chosen to focus my work on the second pillar of the directive and more precisely on the Own Risk and Solvency Assessment (ORSA) process. This regulatory tool is in fact the second major source of complexity in Solvency II. It is a risk management process totally integrated in the company whose objective is to lead insurers to a better understanding of their risks. During my work, I tried to conceptualize and to propose operational implementations to answer the problems induced by the ORSA (calculation of the Global Solvency Requirement and Permanent Compliance). Finally, through a joint work with N. El Karoui, S. Loisel and J.-L. Prigent, we analyzed and exemplified some of the major dangers induced by economic valuation.

In this paper, we introduce a new structured financial product: the so-called Life Nominal Chooser Swaption (LNCS). Thanks to such a contract, insurers could keep pure longevity risk and transfer a great part of interest rate risk underlying annuity portfolios to financial markets. Before the issuance of the contract, the insurer determines a confidence band of survival curves for her portfolio. An interest rate hedge is set up, based on swaption mechanisms. The bank uses this band as well as an interest rate model to price the product. At the end of the first period (e.g. 8 to 10 years), the insurer has the right to enter into an interest rate swap with the bank, where the nominal is adjusted to her (re-forecasted) needs. She chooses (inside the band) the survival curve that better fits her anticipation of future mortality of her portfolio (during 15 to 20 more years, say) given the information available at that time. We use a population dynamics longevity model and a classical two-factor interest rate model %two-factor Heath-Jarrow-Morton (HJM) model for interest rates to price this product. Numerical results show that the option offered to the insurer (in terms of choice of nominal) is not too expensive in many real-world cases. We also discuss the pros and the cons of the product and of our methodology. This structure enables insurers and financial institutions to remain in their initial field of expertise.

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Landfill bioreactors are based on an acceleration of in-situ waste biodegradation by performing leachate recirculation. To quantify the water content and to evaluate the leachate injection system, in-situ methods are required to obtain spatially distributed information, usually electrical resistivity tomography (ERT). In a previous study, the MICS (multiple inversions and clustering strategy) methodology was proposed to improve the hydrodynamic interpretation of ERT results by a precise delimitation of the infiltration area. In this study, MICS was applied on two ERT time-lapse data sets recorded on different waste deposit cells in order to compare the hydrodynamic behaviour of leachate flow between the two cells. This comparison is based on an analysis of: (i) the volume of wetted waste assessed by MICS and the wetting rate, (ii) the infiltration shapes and (iii) the pore volume used by the leachate flow. This paper shows that leachate hydrodynamic behaviour is comparable from one waste deposit cell to another with: (i) a high leachate infiltration speed at the beginning of the infiltration, which decreases with time, (ii) a horizontal anisotropy of the leachate infiltration shape and (iii) a very small fraction of the pore volume used by the leachate flow. This hydrodynamic information derived from MICS results can be useful for subsurface flow modelling used to predict leachate flow at the landfill scale.

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A numerical method to approximate ruin probabilities is proposed within the frame of a compound Poisson ruin model. The defective density function associated to the ruin probability is projected in an orthogonal polynomial system. These polynomials are orthogonal with respect to a probability measure that belongs to Natural Exponential Family with Quadratic Variance Function (NEF-QVF). The method is convenient in at least four ways. Firstly, it leads to a simple analytical expression of the ultimate ruin probability. Secondly, the implementation does not require strong computer skills. Thirdly, our approximation method does not necessitate any preliminary discretisation step of the claim sizes distribution. Finally, the coefficients of our formula do not depend on initial reserves.

Modeling and forecasting damage from wind storms is a major issue for insurance companies. In this article, we focus on the sensitivity of estimations of return periods for extreme events with respect to modeling assumptions and the type of input data. Numerous variables play a role: the quality of data concerning the location of insured buildings and weather report homogeneity, missing updates for correcting non-stationarities concerning the insurance portfolio history, ground roughness or climate change, the evolution of the model after an unprecedented event such as the Lothar storm observed in 1999 in Europe, temporal aggregation of daily events over several days, where events could span over several days up to one week, and storm trajectories, which could change due to global warming or sweep larger areas. Our work explores three important aspects. First, we highlight the geographic heterogeneity of the spatial distribution of wind speeds and the resulting damages. Therefore, we propose to partition the French territory into 6 relatively homogeneous storm zones, based on the dependence among observed wind speeds and geographic distance. Second, we extend a storm index—defined in Mornet et al. (Risk Anal 35:2029–2056, 2015)—to take into account geographic heterogeneity, and we analyze its tail behavior to show the difficulties met to obtain reliable results on extreme events. Third, we explore the calculation of Solvency Capital Requirements based on a model that we propose for the annual claim amount distribution. The purpose of our analysis is to quantify and to point out the high level of uncertainty in the computation of return periods and of other quantities strongly influenced by extreme events.

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Models and forecasts of damage from wind storms are a major issue for insurance companies. In this article, we focus on the calculation sensitivity of return periods for extreme events. Numerous elements come into play, such as data quality (location of insured buildings, weather report homogeneity), missing updates (history of insurance portfolios, change of ground roughness, climate change), the evolution of the model after an unprecedented event such as Lothar in Europe and temporal aggregation (events defined through blocks of 2 or 3 days or blocks of one week). Another important aspect concerns storm trajectories, which could change due to global warming or sweep larger areas. We here partition the French territory into 6 storm zones depending on extreme wind correlation to test several scenarios. We use a storm index defined in \cite{Ma} to show the difficulties met to obtain reliable results on extreme events.

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Most mortality models are generally calibrated on national population. However, pensions funds and annuity providers are mainly interested in the mortality rates of their own portfolio. In this paper we put forward a multivariate approach for forecasting pairwise mortality rates of related population. The investigated approach links national population mortality to a subset population using an econometric model that captures a long-run relationship between both mortality dynamics. This model does not lay the emphasis on the correlation that the two given mortality dynamics would present but rather on the long-term behaviour, which suggests that the two time-series cannot wander off in opposite directions for very long without mean reverting force on grounds of biological reasonableness. The model additionally captures the short-run adjustment between the considered mortality dynamics. Our aim is to propose a consistent approach to forecast pairwise mortality and to some extent to better control and assess basis risk underlying index-based longevity securitization. An empirical comparison of the forecast of one-year death probabilities of portfolio-experienced mortality is performed using both a factor-based model and the proposed approach. The robustness of the model is tested on mortality rate data for England & Wales and Continuous Mortality Investigation assured lives representing a sub-population.

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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.

This is a summary of the main topics and findings from the Swiss Risk and Insurance Forum 2015. That event gathered experts from academia, insurance industry, regulatory bodies, and consulting companies to discuss the past and current developments and necessary next steps for dealing with old-age provision. Topics include the pension funding gap, demographic and societal challenges, the valuation of pension liabilities, economic and regulatory capital models, and the role of financial markets.

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.

In this paper, our aim is to measure mortality rates which are specific to individual observable factors when these can change during life. The study is based on longitudinal data recording marital status and socio-professional features at census times, therefore the observation scheme is interval-censored since individual characteristics are only observed at isolated dates and transition times remain unknown. To this aim, we develop a parametric maximum likelihood estimation procedure for multi-state models that takes into account both interval-censoring and reversible transitions. This method, inspired by recent advances in the statistical literature, allows us to capture characteristic-specific mortality rates, in particular to recover the mortality compensation law at high ages, but also to capture the age pattern of characteristics changes. The dynamics of several population compositions is addressed, and allows us to give explanations on the pattern of aggregate mortality, as well as on the impact on typical life insurance products. Particular attention is devoted to characteristics changes and parameter uncertainty that are both crucial to take into account.

For insurance companies, wind storms represent a main source of volatility, leading to potentially huge aggregated claim amounts. In this article, we compare different constructions of a storm index allowing us to assess the economic impact of storms on an insurance portfolio by exploiting information from historical wind speed data. Contrary to historical insurance portfolio data, meteorological variables can be considered as stationary between years and are easily available with long observation records. hence, they represent a valuable source of additional information for insurers if the relation between observations of claims and wind speeds can be revealed. Since standard correlation measures between raw wind speeds and insurance claims are weak, a storm index focusing on high wind speeds can afford better information. This method has been used on the German territory by Klawa and Ulbrich and gave good results for yearly aggregated claims. Using historical meteorological and insurance data, we assess the consistency of the pro-posed indices construction and we test their sensitivity to their various parameters and weights. Moreover, we are able to place the major insurance events since 1998 on a broader horizon of 40+ years. Our approach provides a meteorological justification for calculating the return periods of extreme storm-related insurance events whose magnitude has rarely been reached.

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This paper introduces a class of Schur-constant survival models, of dimension n, for arithmetic non-negative random variables. Such a model is defined through a univariate survival function that is shown to be n-monotone. Two general representations are obtained, by conditioning on the sum of the n variables or through a doubly mixed multinomial distribution. Several other properties including correlation measures are derived. Three processes in insurance theory are discussed for which the claim interarrival periods form a Schur-constant model.

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In the present paper we develop recursive algorithms to evaluate the distribution of the net present value (abbreviated as "NPV") of a health care contract. The duration of the program is a random variable representing the lifetime of an individual. We suggest a discrete time phase-type approach to model individual health care costs. In this approach, annual health care costs depend naturally on the health state of the individual. We also derive the distribution of the NPV assuming that annual health care costs are iid random variables. We demonstrate analytically that, under special parametrisation, the model with iid costs gives a similar expectation of the NPV to the one of the model with health dependent costs. We propose techniques to evaluate the impact of health related events and demonstrate it on numerical examples.

Motivated by seasonality and regime-switching features of some insurance claim counting processes, we study the statistical analysis of a Markov-modulated Poisson process featuring seasonality. We prove the strong consistency and the asymptotic normality of a maximum split-time likelihood estimator of the parameters of this model, and present an algorithm to compute it in practice. The method is illustrated on a small simulation study and a real data analysis.

For insurance companies, wind storms represent a main source of volatility, leading to potentially huge aggregated claim amounts. In this article, we compare different constructions of a storm index allowing us to assess the economic impact of storms on an insurance portfolio by exploiting information from historical wind speed data. Contrary to historical insurance portfolio data, meteorological variables show fewer nonstationarities between years and are easily available with long observation records. hence, they represent a valuable source of additional information for insurers if the relation between observations of claims and wind speeds can be revealed. Since standard correlation measures between raw wind speeds and insurance claims are weak, a storm index focusing on high wind speeds can afford better information. A storm index approach has been applied to yearly aggregated claim amounts in Germany with promising results. Using historical meteorological and insurance data, we assess the consistency of the proposed index constructions with respect to various parameters and weights. Moreover, we are able to place the major insurance events since 1998 on a broader horizon beyond 40 years. Our approach provides a meteorological justification for calculating the return periods of extreme-storm-related insurance events whose magnitude has rarely been reached.

In risk management, the distribution of underlying random variables is not always known. Sometimes, only the mean value and some shape information (decreasingness, convexity after a certain point,.) of the discrete density are available. The present paper aims at providing convex extrema in some cases that arise in practice in insurance and in other fields. This enables us to obtain for example bounds on variance and on Solvency II related quantities in insurance applications. In this paper, we first consider the class of discrete distributions whose probability mass functions are nonincreasing on a support ${\cal D}_n\equiv \{0,1,\ldots,n\}$. Convex extrema in that class of distributions are well-known. Our purpose is to point out how additional shape constraints of convexity type modify these extrema. Three cases are considered: the p.m.f. is globally convex on $\N$, it is convex only from a given positive point $m$, or it is convex only up to some positive point $m$. The corresponding convex extrema are derived by using simple crossing properties between two distributions. The influence of the choice of $n$ and $m$ is discussed numerically, and several illustrations to ruin problems are presented. These results provide a complement to two recent works by Lefévre and Loisel (2010), (2012).

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A numerical method to compute bivariate probability distributions from their Laplace transforms is presented. The method consists in an orthogonal projection of the probability density function with respect to a probability measure that belongs to a Natural Exponential Family with Quadratic Variance Function (NEF-QVF). A particular link to Lancaster probabilities is highlighted. The procedure allows a quick and accurate calculation of probabilities of interest and does not require strong coding skills. Numerical illustrations and comparisons with other methods are provided. This work is motivated by actuarial applications. We aim at recovering the joint distribution of two aggregate claims amounts associated with two insurance policy portfolios that are closely related, and at computing survival functions for reinsurance losses in presence of two non-proportional reinsurance treaties.

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Worldwide, local anthropogenic extinctions have recently been reported to induce trophic cascades, defined as perturbations of top consumers that propagate along food chains down to primary producers. This focus on the effects of top-consumer extinction (i.e. of species presence) ignores potential cascading effects of the rapid morphological changes that may precede extinction. Here, we show in an experimental, three-level food chain including medaka fish, herbivorous zooplankton and unicellular algae that varying body length of a single fish from large (36.3 mm) to small (11.5 mm) induced a stronger trophic cascade than varying an average-sized (23.8 mm) fish from being present to absent. The strength of fish predation on zooplankton scaled quasi linearly (not with a power exponent) with fish body length and associated gape width, suggesting that the resultant trophic cascade was morphology (not metabolism)-dependent. The effect of fish body length was stronger on phyto-than on zooplankton, because large-sized fish had the unique ability to suppress large-sized herbivores, which in turn had high grazing capacities. Hence, our results show that consumer body size, by setting diet breadth, can both drive and magnify the strength of trophic cascades. In contrast, fish body shape had no significant effect on fish predatory performances when its allometric component (the effect of size on shape) was removed. In the wild, human-induced body downsizing of top consumers is widespread, and mitigating the resultant perturbations to ecosystem function and services will require a paradigm shift from preserving species presence towards preserving species size structure.

In this paper, we obtain asymptotic ruin probabilities in two models where claim amounts become more and more adverse, because of phenomena like climate change or some kind of sectorial inflation. The method we use also enables us to study a risk model in which claims have infinite mean. In such models, ruin probability can be controlled by a strong increase in the premium income rate, which causes premium to become unacceptable for customers. We provide numerical illustrations of the impact of the (uncertain) speed of change in the parameter of the claim size distribution, both in terms of ruin and in terms of time at which premium becomes too high.

In the present paper we develop recursive algorithms to evaluate the distribution of the net present value (abbreviated as "NPV") of a health care contract. The duration of the program is a random variable representing the lifetime of an individual. We suggest a discrete time phase-type approach to model individual health care costs. In this approach, annual health care costs depend naturally on the health state of the individual. We also derive the distribution of the NPV assuming that annual health care costs are iid random variables. We demonstrate analytically that, under special parametrisation, the model with iid costs gives a similar expectation of the NPV to the one of the model with health dependent costs. We propose techniques to evaluate the impact of health related events and demonstrate it on numerical examples.

We consider the minimax quickest detection problem of an unobservable time of change in the rate of an inhomogeneous Poisson process. We seek a stopping rule that minimizes the robust Lorden criterion, formulated in terms of the number of events until detection, both for the worst-case delay and the false alarm constraint. In the Wiener case, such a problem has been solved using the so-called cumulative sums (cusum) strategy by Shiryaev [33, 35], or Moustakides [24] among others. In our setting, we derive the exact optimality of the cusum stopping rule by using finite variation calculus and elementary martingale properties to characterize the performance functions of the cusum stopping rule in terms of scale functions. These are solutions of some delayed differential equations that we solve elementarily. The case of detecting a decrease in the intensity is easy to study because the performance functions are continuous. In the case of an increase where the performance functions are not continuous, martingale properties require using a discontinuous local time. Nevertheless, from an identity relating the scale functions, the optimality of the cusum rule still holds. Finally, some numerical illustration are provided.

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In the automobile insurance sector, the decision of the Court of Justice of the European Union that it is no longer possible to charge rates according to the gender of the insured, as well as the spread of aukilo insurance schemes, necessarily lead to an evolution of risk management. In this article, we use the Allianz portfolio in France to address these two issues. We propose to characterize the male/female partition by exploring different statistical methods such as the logistic procedure, the multiple correspondence analysis (MCA) or the classification trees (CART). We show that it is possible to compensate for the absence of the sex variable by other information specific to the insured or to his vehicle and in particular the use of mileage records [11]. We then return to the use of generalized linear models (GLM) to validate these results. In a second part, we focus on the experience of novice drivers. This category of insureds is among the most sensitive to male/female criteria in its pricing. Here, Generalized Additive Models (GAM) are used to exploit numerical variables such as the annual mileage driven. Finally, we propose to study the on-road behavior of the insured during his three years of novitiate to create new risk categories.

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As more and more people believe that significant life extensions may come soon, should commonly used future mortality assumptions be considered prudent? We find here that commonly used actuarial tables for annuitants – as well as the Lee-Carter model – do not extrapolate life expectancy at the same rate for future years as for past years. instead they produce some longevity deceleration. This is typically because their mortality improvements decrease after a certain age, and those age-specific improvements are constant over time. As potential alternatives i) we study the Bongaarts model that produces straight increases in life expectancy. ii) we adapt it to produce best-practice longevity trends iii) we compare with various longevity scenarios even including a model for “life extension velocity”. iv) after gathering advances in biogerontology we discuss elements to help retirement systems resist to a potential strong increase in life expectancy.

We use a brand new data-set built from French supervisory reports to investigate the drivers of the participation rates served on euro-denominated life-insurance contracts over the period 1999-2013. Our analysis confirms practitioners’ insight on the alignment with the 10-years French government bond, yet we show that on aggregate, insurers serve less than this target. Our data indicate that financial margins are more strictly targeted than participation. We find evidence that lapses are fairly uncorrelated with participation, suggesting other levers to pilot surrenders. If higher asset returns can imply better yield for policyholders, riskier portfolios do not translate into better participation.

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In this thesis, we study storm coverage dedicated to wind damage and a development of behavioral insurance through automobile risk. We combine external information such as wind speed with insurance data. We propose the construction of a storm index to complete and reinforce the evaluation of damages caused by major storms. We then define a division of the French territory into 6 storm zones, depending on the extreme wind correlations, to test several scenarios. These different tests and considerations allow us to improve our storm index. We use extreme value theory models to show the impact of variability on the calculation of return periods and capital requirements. We thus highlight the difficulties encountered in obtaining robust results in relation to extreme events. In the case of motor insurance, we test different methods to respond to technical and regulatory changes. We characterize the male/female partition using the logistic procedure, multiple correspondence analysis or classification trees. We show that it is possible to compensate for the absence of the gender variable by other information specific to the insured or to his vehicle and in particular the use of mileage records. Finally, we are interested in the experience acquired by novice drivers. We study the on-road behavior of the insured to create new risk classes.

The storage of waste in a non-hazardous final waste storage facility (ISDUND) is an essential link in the management of our waste flows. Although organic matter recovery and recycling processes are developing, there is still a significant proportion of biodegradable waste in these waste streams. Bioactive ISDUNDs, also called bioreactors, are an alternative to "classic" ISDUNDs. In these installations, the leachate recovered from the bottom of the cell is generally reinjected at the top of the waste mass to percolate by gravity. The objective of the increase in humidity thus obtained is to accelerate the degradation of the waste, which remains one of the key parameters for obtaining optimal anaerobic biodegradation of the organic matter of the waste in ISDUND. The effect sought by the operators is both to increase the quantity of biogas produced and to concentrate its production over a shorter period of time in order to allow an easier recovery of the latter. These objectives also lead to an important issue which is the faster and more homogeneous stabilization of the waste mass. However, at the present time, few diagnostic tools allow to inform on the possible presence, in the waste mass, of more or less degraded zones. The use of these tools is also part of the problem of the end of the post-exploitation follow-up, which is fixed by law at a minimum of 30 years. The lack of parameters or quantitative data on these systems is an obstacle to the optimization of this management method. As part of a multi-partner research project, a set of physical measurement methods providing information on variations in water content and temperature is being deployed on bioactive ISDUND cells and combined with laboratory tests to optimize the cycles of reinjection sequences envisaged in situ. Started at the end of 2011, the instrumentation is now in its final phase of implementation on site. This article proposes to illustrate the methods and the approach implemented as well as to present the first results obtained during the on-site measurements, in particular concerning the evolution of the temperature and the electrical resistivity measurements.

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In this paper, we obtain asymptotic ruin probabilities in two models where claim amounts become more and more adverse, because of phenomena like climate change or some kind of sectorial inflation. The method we use also enables us to study a risk model in which claims have infinite mean. In such models, ruin probability can be controlled by a strong increase in the premium income rate, which causes premium to become unacceptable for customers. We provide numerical illustrations of the impact of the (uncertain) speed of change in the parameter of the claim size distribution, both in terms of ruin and in terms of time at which premium becomes too high.

Motivated by seasonality and regime-switching features of some insurance claim counting processes, we study the statistical analysis of a Markov-modulated Poisson process featuring seasonality. We prove the strong consistency and the asymptotic normality of a maximum split-time likelihood estimator of the parameters of this model, and present an algorithm to compute it in practice. The method is illustrated on a small simulation study and a real data analysis.

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In this paper, we propose some characteristics of next-year impairments in a generic Black & Scholes framework, with one equity security, and under IFRS rules. We derive expression for the probability of impairment event for an equity-security recognized in the available-for-sale (AFS) category. Our decomposition of this event is also useful to retrieve barrier options valuation methods. From there, we obtain an explicit formula for the rst moment of impairment value and its cumulative distribution function, as well as sensitivities. Numerical studies are carried out on concrete securities. We also study a mean-preserving one-criterion proxy used by some insurance practitioners for the next-year impairment losses and discuss its relevance. More generally, our study paves the way for applications of nancial mathematics techniques to accounting issues related to impairments in the IFRS framework.

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This paper studies a new risk measure derived from the expected area in red introduced in Loisel (2005). Specifically, we derive various properties of a risk measure defined as the smallest initial capital needed to ensure that the expected time-integrated negative part of the risk process on a fixed time interval [0. T] (T can be infinite) is less than a given predetermined risk limit. We also investigate the optimal risk limit allocation: given a risk limit set at company level for the sum of the expected areas in red of all lines, we determine the way(s) to allocate this risk limit to the subsequent business lines in order to minimize the overall capital needs.

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The goal of this paper is to give recent results in risk theory presented at the Conference "Journée MAS 2012" which took place in Clermont Ferrand. After a brief state of the art on ruin theory, we explore some particular aspects and recent results. One presents matrix exponential approximations of the ruin probability. Then we present asymptotics of the ruin probability based on mixing properties of the claims distribution. Finally, the multivariate case, motivated by reinsurance, is presented and some contemporary results (closed forms and asymptotics) are given.

This paper studies a new risk measure derived from the expected area in red introduced in Loisel (2005). Specifically, we derive various properties of a risk measure defined as the smallest initial capital needed to ensure that the expected time-integrated negative part of the risk process on a fixed time interval [0. T] (T can be infinite) is less than a given predetermined risk limit. We also investigate the optimal risk limit allocation: given a risk limit set at company level for the sum of the expected areas in red of all lines, we determine the way(s) to allocate this risk limit to the subsequent business lines in order to minimize the overall capital needs.

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 order to produce more efficiently renewable energy in landfill from the municipal solid waste by biogas production and to enhance the degradation of the organic fraction of waste, landfill bioreactor using leachate recirculation is a promising management strategy. Bioreactors are based on an acceleration of in-situ waste biodegradation by reaching biochemical optimal moisture content with performing leachate injection Thus, a key need is to obtain a better understanding of leachate infiltration into the waste mass during recirculation events, but also of biodegradation process at waste deposit cell scale. During “Champs Jouault” research project, for the first time, combination of different methods of measurements is managed in the same place at field scale: Electrical Resistivity Tomography (ERT) for leachate diffusion. fiber optic lines for distributed temperature measurements for biodegradation study. Dielectric permittivity with Time domain transmission as indicator of moisture variations. Two time of sample recording are proposed: a short-monitoring to evaluate leachate flow (recirculation device) in waste mass and a long-term monitoring to study biodegradation process.

This paper is concerned with the class of distributions, continuous or discrete, whose shape is monotone of finite integer order t. A characterization is presented as a mixture of a minimum of t independent uniform distributions. Then, a comparison of t-monotone distributions is made using the s-convex stochastic orders. A link is also pointed out with an alternative approach to monotonicity based on a stationary-excess operator. Finally, the monotonicity property is exploited to reinforce the classical Markov and Lyapunov inequalities. The results are illustrated by several applications to insurance.

In risk management, the distribution of underlying random variables is not always known. Sometimes, only the mean value and some shape information (decreasingness, convexity after a certain point,.) of the discrete density are available. The present paper aims at providing convex extrema in some cases that arise in practice in insurance and in other fields. This enables us to obtain for example bounds on variance and on Solvency II related quantities in insurance applications. In this paper, we first consider the class of discrete distributions whose probability mass functions are nonincreasing on a support ${\cal D}_n\equiv \{0,1,\ldots,n\}$. Convex extrema in that class of distributions are well-known. Our purpose is to point out how additional shape constraints of convexity type modify these extrema. Three cases are considered: the p.m.f. is globally convex on $\N$, it is convex only from a given positive point $m$, or it is convex only up to some positive point $m$. The corresponding convex extrema are derived by using simple crossing properties between two distributions. The influence of the choice of $n$ and $m$ is discussed numerically, and several illustrations to ruin problems are presented. These results provide a complement to two recent works by Lefévre and Loisel (2010), (2012).

We study a new risk measure inspired from risk theory with a heat wave risk analysis motivation. We show that this risk measure and its sensitivities can be computed in practice for relevant temperature stochastic processes. This is in particular useful for measuring the potential impact of climate change on heat wave risk. Numerical illustrations are given.

Market cycles play a great role in reinsurance. Cycle transitions are not independent from the claim arrival process : a large claim or a high number of claims may accelerate cycle transitions. To take this into account, a semi-Markovian risk model is proposed and analyzed. A refined Erlangization method is developed to compute the finite-time ruin probability of a reinsurance company. As this model needs the claim amounts to be Phase-type distributed, we explain how to fit mixtures of Erlang distributions to long-tailed distributions. Numerical applications and comparisons to results obtained from simulation methods are given. The impact of dependency between claim amounts and phase changes is studied.

Industrial operators need spatial measurements to manage their bioreactor landfill if they want to reach a homogeneous bio-mechanical degradation of the waste mass and a biogas production in accordance to the biochemical methane potential evaluated. Electrical resistivity tomography is non intrusive and allows 2D or 3D description of waste mass electrical resistivity. Volumetric water content and temperature are known to influence waste electrical resistivity. For the first time in France, a same waste cell deposit was instrumented with electrical resistivity tomography device, temperature sensors and volumetric water content probes. The objective was to increase confidence on resistivity interpretation. The wide range of temperature recorded demonstrates its influence when waste resistivity has to be compared. Moisture content variations were too weak to explain the resistivity evolution observed.

We present a new model of loss processes in insurance. The process is a couple $(N, \, L)$ where $N$ is a univariate Markov-modulated Poisson process (MMPP) and $L$ is a multivariate loss process whose behaviour is driven by $N$. We prove the strong consistency of the maximum likelihood estimator of the parameters of this model, and present an EM algorithm to compute it in practice. The method is illustrated with simulations and real sets of insurance data.

This paper is concerned with the class of distributions, continuous or discrete, whose shape is monotone of finite integer order t. A characterization is presented as a mixture of a minimum of t independent uniform distributions. Then, a comparison of t-monotone distributions is made using the s-convex stochastic orders. A link is also pointed out with an alternative approach to monotonicity based on a stationary-excess operator. Finally, the monotonicity property is exploited to reinforce the classical Markov and Lyapunov inequalities. The results are illustrated by several applications to insurance.

In this paper, we formulate a noncooperative game to model a non-life insurance market. The aim is to analyze the e ects of competition between insurers through di erent indicators: the market premium, the solvency level, the market share and the underwriting results. Resulting premium Nash equilibria are discussed and numerically illustrated.

The purpose of this paper is to point out that an asymptotic rule "A+B/u" for the ultimate ruin probability applies to a wide class of dependent risk models, in discrete and continuous time. Dependence is incorporated through a mixing approach among claim amounts or claim inter-arrival times, leading to a systemic risk behavior. Ruin corresponds here either to classical ruin, or to stopping the activity after realizing that it is not pro table at all, when one has little possibility to increase premium income rate. Several special cases for which closed formulas are derived, are also investigated in some detail.

In this paper, we consider a discrete-time ruin model where experience rating is taken into account. The main objective is to determine the behavior of the ultimate ruin probabilities for large initial capital in the case of light-tailed claim amounts. The logarithmic asymptotic behavior of the ultimate ruin probability is derived. Typical pathes leading to ruin are studied. An upper bound is derived on the ultimate ruin probability in some particular case. The influence of the number of data points taken into account is analyzed, and numerical illustrations support the theoretical findings. Finally, we investigate the heavy-tailed case. The impact of the number of data points used for the premium calculation appears to be rather different from the one in the light-tailed case.

We formulate a noncooperative game to model competition for policyholders among non-life insurance companies, taking into account market premium, solvency level, market share and underwriting results. We study Nash equilibria and Stackelberg equilibria for the premium levels, and give numerical illustrations.

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.

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.

This document presents a synthesis of my work on mathematical problems applied to actuarial science. The theory of risk, also known as the theory of ruin, concerns in a general way the evaluation of probabilities of realizations of unfavorable events for insurance companies. Beyond these calculations of probabilities in a fixed model, branches of this theory are also interested in various optimization problems: allocation of reserves, dividend or taxation strategy (in the sense of taxation), investment in risky assets, reinsurance program.

The future solvency standards for the insurance industry, Solvency 2, aim to improve risk management by identifying different risk classes and modules, and by allowing companies to use internal models to estimate their regulatory capital. The standard formula defines this capital as being equal to a 99.5% VaR over a one-year horizon for each risk module. Then, at each intermediate consolidation level, the different VaRs are aggregated through a correlation matrix. Several problems appear with this method: - The regulator uses the term "VaR" without communicating marginal or global distributions. This multi-variate risk measure is only relevant if each risk follows a normal distribution. - The time horizon of one year does not correspond to that of an insurance company's commitments, and poses problems when it comes to determining the frequency of updates of internal models. - The dependency structure proposed by the standard formula does not correspond to the one usually implemented by companies and is difficult to use in an internal model. The first part of this paper will present in detail the key points of the reform and will give some thoughts on its application in risk management. In the second part, it will be shown that this multi-variate risk measure does not satisfy the main axioms of a risk measure. Moreover, it does not allow for the comparison of capital requirements between companies, since it is not universal. The third part will show that in order to value capital at an intermediate point before maturity, a risk measure must be able to adjust to different periods, and thus be multi-period. Finally, the fourth part will focus on an alternative to the correlation matrix to model dependence, namely copulas.

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.