RULLIERE Didier

The generalized gamma convolution class of distribution appeared in Thorin's work while looking for the infinite divisibility of the log-Normal and Pareto distributions. Although these distributions have been extensively studied in the univariate case, the multivariate case and the dependence structures that can arise from it have received little interest in the literature. Furthermore, only one projection procedure for the univariate case was recently constructed, and no estimation procedure are available. By expending the densities of multivariate generalized gamma convolutions into a tensorized Laguerre basis, we bridge the gap and provide performant estimations procedures for both the univariate and multivariate cases. We provide some insights about performance of these procedures, and a convergent series for the density of multivariate gamma convolutions, which is shown to be more stable than Moschopoulos's and Mathai's univariate series. We furthermore discuss some examples.

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.

This paper provides an alternative proof of the characterization of multiply monotone functions as integrals of simple polynomial-type applications with respect to a probability measure. This constitutes an analogue of the Bernstein-Widder representation of completely monotone functions as Laplace transforms. The proof given here relies on the abstract representation result of Choquet rather than the analytic derivation originally given by Williamson. To this end, we identify the extreme points in the convex set of multiply monotone functions. Our result thus gives a geometric perspective to Williamson's representation.

For a given random sample from some underlying multivariate distribution F we consider the domination of the component-wise maxima by some independent random vector W with underlying distribution function G. We show that the probability that certain components of the sample maxima are dominated by the corresponding components of W can be approximated under the assumptions that both F and G are in the max-domain of attraction of some max-stable distribution function F and G, respectively. We study further some basic properties of the dominated components of sample maxima by W .

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

In this paper, we investigate the link between the joint law of a d-dimensional random vector and the law of some of its multivariate marginals. We introduce and focus on a class of distributions, that we call projective, for which we give detailed properties. This allows us to obtain necessary conditions for a given construction to be projective. We illustrate our results by proposing some theoretical projective distributions, as elliptical distributions or a new class of distribution having given bivariate margins. In the case where the data do not necessarily correspond to a projective distribution, we also explain how to build proper distributions while checking that the distance to the prescribed projections is small enough.

For a given random sample from some underlying multivariate distribution F we consider the domination of the component-wise maxima by some independent random vector W with underlying distribution function G. We show that the probability that certain components of the sample maxima are dominated by the corresponding components of W can be approximated under the assumptions that both F and G are in the max-domain of attraction of some max-stable distribution function F and G, respectively. We study further some basic properties of the dominated components of sample maxima by W .

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

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The class of multivariate Archimedean copulas is defined by using a real-valued function called the generator of the copula. This generator satisfies some properties, including d-monotony. We propose here a new basic transformation of this generator, preserving these properties, thus ensuring the validity of the transformed generator and inducing a proper valid copula. This transformation acts only on a specific portion of the generator, it allows both the non-reduction of the likelihood on a given dataset, and the choice of the upper tail dependence coefficient of the transformed copula. Numerical illustrations show the utility of this construction, which can improve the fit of a given copula both on its central part and its tail.

This paper collects the contributions which were presented during the session devoted to Gaussian processes at the Journées MAS 2016. First, an introduction to Gaussian processes is provided, and some current research questions are discussed. Then, an application of Gaussian process modeling under linear inequality constraints to financial data is presented. Also, an original procedure for handling large data sets is described. Finally, the case of Gaussian process based iterative optimization is discussed.

Kriging is a widely employed technique, in particular for computer experiments, in machine learning or in geostatistics. An important challenge for Kriging is the computational burden when the data set is large. We focus on a class of methods aiming at decreasing this computational cost, consisting in aggregating Kriging predictors based on smaller data subsets. We prove that aggregations based solely on the conditional variances provided by the different Kriging predictors can yield an inconsistent final Kriging prediction. In contrasts, we study theoretically the recent proposal by [Rullière et al., 2017] and obtain additional attractive properties for it. We prove that this predictor is consistent, we show that it can be interpreted as an exact conditional distribution for a modified process and we provide error bounds for it.

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This work falls within the context of predicting the value of a real function at some input locations given a limited number of observations of this function. The Kriging interpolation technique (or Gaussian process regression) is often considered to tackle such a problem, but the method suffers from its computational burden when the number of observation points is large. We introduce in this article nested Kriging predictors which are constructed by aggregating sub-models based on subsets of observation points. This approach is proven to have better theoretical properties than other aggregation methods that can be found in the literature. Contrarily to some other methods it can be shown that the proposed aggregation method is consistent. Finally, the practical interest of the proposed method is illustrated on simulated datasets and on an industrial test case with (Formula presented.) observations in a 6-dimensional space.

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

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

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

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

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

This paper presents the impact of a class of transformations of copulas in their upper and lower multivariate tail dependence coefficients. In particular we focus on multivariate Archimedean copulas. In the first part of this paper, we calculate multivariate tail dependence coefficients when the generator of the considered copula exhibits some regular variation properties, and we investigate the behaviour of these coefficients in cases that are close to tail independence. This first part exploits previous works of Charpentier and Segers (2009) and extends some results of Juri and Wüthrich (2003) and De Luca and Rivieccio (2012). We also introduce a new Regular Index Function (RIF) exhibiting some interesting properties. In the second part of the paper we analyse the impact in the upper and lower multivariate tail dependence coefficients of a large class of transformations of dependence structures. These results are based on the transformations exploited by Di Bernardino and Rullière (2013). We extend some bivariate results of Durante et al. (2010) in a multivariate setting by calculating multivariate tail dependence coefficients for transformed copulas. We obtain new results under specific conditions involving regularly varying hazard rates of components of the transformation. In the third part, we show the utility of using transformed Archimedean copulas, as they permit to build Archimedean generators exhibiting any chosen couple of lower and upper tail dependence coefficients. The interest of such study is also illustrated through applications in bivariate settings. At last, we explain possible applications with Markov chains with specific dependence structure.

An important topic in Quantitative Risk Management concerns the modeling of dependence among risk sources and in this regard Archimedean copulas appear to be very useful. However, they exhibit symmetry, which is not always consistent with patterns observed in real world data. We investigate extensions of the Archimedean copula family that make it possible to deal with asymmetry. Our extension is based on the observation that when applied to the copula the inverse function of the generator of an Archimedean copula can be expressed as a linear form of generator inverses. We propose to add a distortion term to this linear part, which leads to asymmetric copulas. Parameters of this new class of copulas are grouped within a matrix, thus facilitating some usual applications as level curve determination or estimation. Some choices such as sub-model stability help associating each parameter to one bivariate projection of the copula. We also give some admissibility conditions for the considered copulas. We propose different examples as some natural multivariate extensions of Farlie-Gumbel-Morgenstern or Gumbel-Barnett.

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Due to the lack of reliable market information, building financial term-structures may be associated with a significant degree of uncertainty. In this paper, we propose a new term-structure interpolation method that extends classical spline techniques by additionally allowing for quantification of uncertainty. The proposed method is based on a generalization of kriging models with linear equality constraints (market-fit conditions) and shape-preserving conditions such as monotonicity or positivity (no-arbitrage conditions). We define the most likely curve and show how to build confidence bands. The Gaussian process covariance hyper-parameters under the construction constraints are estimated using cross-validation techniques. Based on observed market quotes at different dates, we demonstrate the efficiency of the method by building curves together with confidence intervals for term-structures of OIS discount rates, of zero-coupon swaps rates and of CDS implied default probabilities. We also show how to construct interest-rate surfaces or default probability surfaces by considering time (quotation dates) as an additional dimension.

The construction of term structures is at the heart of financial evaluation and risk management see e.g. [1], [2], [3], [4] and [5]. A term structure is a curve that describes the evolution of an economic or financial quantity as a function of the maturity or time horizon. Typical examples are the term structure of risk-free interest rates, the term structure of bonds, the term structure of default probabilities and the term structure of implied volatilities of financial asset returns. In practice, market quotes of the underlying financial products are used and provide partial information on the term structures considered. Moreover, this information is more or less reliable depending on the liquidity of the maturity of the markets in question. The goal is to obtain a continuous maturity curve from this information.

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

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

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

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

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

Given a first set of observations from a design of experiments sampled randomly in the design space, the corresponding set of non-dominated points usually does not give a good approximation of the Pareto front. We propose here to study this problem from the point of view of multivariate analysis, introducing a probabilistic framework with the use of copulas. This approach enables the expression of level lines in the objective space, giving an estimation of the position of the Pareto front when the level tends to zero. In particular, when it is possible to use Archimedean copulas, analytical expressions for Pareto front estimators are available. Several case studies illustrate the interest of the approach, which can be used at the beginning of the optimization when sampling randomly in the design space.

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Calculating return periods and critical layers (i.e., multivariate quantile curves) in a multivariate environment is a di cult problem. A possible consistent theoretical framework for the calculation of the return period, in a multi-dimensional environment, is essentially based on the notion of copula and level sets of the multivariate probability distribution. In this paper we propose a fast and parametric methodology to estimate the multivariate critical layers of a distribution and its associated return periods. The model is based on transformations of the marginal distributions and transformations of the dependence structure within the class of Archimedean copulas. The model has a tunable number of parameters, and we show that it is possible to get a competitive estimation without any global optimum research. We also get parametric expressions for the critical layers and return periods. The methodology is illustrated on hydrological 5-dimensional real data. On this real data-set we obtain a good quality of estimation and we compare the obtained results with some classical parametric competitors.

It is common in optimization to start with a random draw in the space of variables to initialize a population or create a metamodel. In particular, in the multi-objective case, this leads to a set of non-dominated p oints that only inform p eu about the true Pareto front. We propose to study this problem from the point of view of multivariate analysis, by introducing a probabilistic framework and in particular by using copulas. Thus, expressions for the level lines are available in the objective space and consequently allow to obtain an estimate of the position of the Pareto front, when the level tends to zero. Explicit analytical expressions are available when Archimedean copulas are used. The corresponding estimation procedure is detailed and then applied on several examples.

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

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Let $X$ be a mixed process, sum of a brownian motion and a renewal-reward process, and $\tau_{x}$ be the first passage time of a fixed level $x<0$ by $X$. We prove that $\tau_x$ has a density and we give a formula for it. Links with ruin theory are presented. Our result may be computed in classical settings (for a Lévy or Sparre Andersen process) and also in a non markovian context with possible positive and negative jumps. Some numerical applications illustrate the interest of this density formula.

In this paper, we propose a parametric model for multivariate distributions. The model is based on distortion functions, i.e. some transformations of a multivariate distribution which permit to generate new families of multivariate distribution functions. We derive some properties of considered distortions. A suitable proximity indicator between level curves is introduced in order to evaluate the quality of candidate distortion parameters. Using this proximity indicator and properties of distorted level curves, we give a speci c estimation procedure. The estimation algorithm is mainly relying on straightforward univariate optimizations, and we nally get parametric representations of both multivariate distribution functions and associated level curves. Our results are motivated by applications in multivariate risk theory. The methodology is illustrated on simulated and real examples.

We consider the problem of the global minimization of a function observed with noise. This problem occurs for example when the objective function is estimated through stochastic simulations. We propose an original method for iteratively partitioning the search domain when this area is a nite union of simplexes. On each subdomain of the partition, we compute an indicator measuring if the subdomain is likely or not to contain a global minimizer. Next areas to be explored are chosen in accordance with this indicator. Con dence sets for minimizers are given. Numerical applications show empirical convergence results, and illustrate the compromise to be made between the global exploration of the search domain and the focalization around potential minimizers of the problem.

We study the impact of certain transformations within the class of Archimedean copulas. We give some admissibility conditions for these transformations, and define some equivalence classes for both transformations and generators of Archimedean copulas. We extend the $r$-fold composition of the diagonal section of a copula, from $r \in \N$ to $r \in \R$. This extension, coupled with results on equivalence classes, gives us new expressions of transformations and generators. Estimators deriving directly from these expressions are proposed and their convergence is investigated. We provide confidence bands for the estimated generators. Numerical illustrations show the empirical performance of these estimators.

In this thesis, we propose to study the longevity risk and its impact on insurers and pension plan providers. In the first part, we focus on the different mortality models. Based on this literature review, we propose an extension of the so-called CBD model, using Lévy processes, which will take into account the effects of jumps in the mortality curve. In the second part of the thesis, this new model will be used as a basis for the valuation of longevity derivatives. We use as valuation measures the so-called Wang and Esscher transforms that we will have previously defined and justified as being martingale measures equivalent to the historical measure. Finally, we propose a new contract called "mortality collar" which, by its definition, allows an efficient hedge against longevity/mortality risk, both for an insurer and for a pension fund. We provide an in-depth analysis of this risk management tool, both in its mechanism and in its valuation.

The present paper provides a multi-period contagion model in the credit risk field. Our model is an extension of Davis and Lo's infectious default model. We consider an economy of n firms which may default directly or may be infected by other defaulting firms (a domino effect being also possible). The spontaneous default without external influence and the infections are described by not necessarily independent Bernoulli-type random variables. Moreover, several contaminations could be required to infect another firm. In this paper we compute the probability distribution function of the total number of defaults in a dependency context. We also give a simple recursive algorithm to compute this distribution in an exchangeability context. Numerical applications illustrate the impact of exchangeability among direct defaults and among contaminations, on different indicators calculated from the law of the total number of defaults. We then examine the calibration of the model on iTraxx data before and during the crisis. The dynamic feature together with the contagion effect seem to have a significant impact on the model performance, especially during the recent distressed period.

This thesis focuses on strategic asset allocation models and their application to the management of financial reserves in pay-as-you-go pension plans, particularly those that are partially funded. The study of the usefulness of reserves for a pay-as-you-go system and a fortiori their management remains a little explored subject. Conventional assumptions are sometimes considered too restrictive to describe the complex evolution of reserves. New models and results are developed at three levels: the generation of economic scenarios (GSE), numerical optimization techniques and the choice of the optimal strategic allocation in an asset-liability management (ALM) context. Within the framework of economic and financial scenario generation, some performance measurement indicators of the GSE have been studied. In addition, improvements have been made compared to what is usually done in the construction of the GSE, particularly in the choice of the correlation matrix between the modeled variables. Concerning the calibration of the GSE, a set of tools allowing the estimation of its different parameters has been presented. This thesis has also paid particular attention to the numerical techniques for finding the optimum, which remain essential issues for the implementation of an allocation model. A reflection on a global optimization algorithm for a non-convex and noisy function has been developed. The algorithm allows to easily modulate, by means of two parameters, the reiteration of draws in a neighborhood of the discovered solution points, or conversely the exploration of the function in unexplored areas. We then present innovative ALM techniques based on stochastic programming. Their application has been developed for the choice of the strategic asset allocation of partially funded pay-as-you-go pension plans. A new methodology for the generation of the scenario tree was adopted at this level. Finally, a comparative study of the developed ALM model with the one based on the Fixed-Mix strategy was performed. Various sensitivity tests were also carried out to measure the impact of changes in certain key input variables on the results produced by our ALM model.

With the development of the marketing of life insurance products, and in particular risk products, the market participants are led to question the technical bases constituting their activity. Thus, the control of technical risk has become a major challenge for a life insurance company. Even though risk management is at the heart of the insurer's activity, the latter must have the means to meet its commitments. This requirement requires, on the one hand, the analysis of past observations, on the other hand, the forecasting of future commitments, and finally the reaction to these analyses, aiming at establishing a conformity between knowledge and the strategy of its development. In response to these three problems, three parts successively detail the observation, forecasting and control of technical risk, studied as a non-financial cause of discrepancies between forecasts and actual activity. These parts are organized in order to gradually summarize the information available to a life insurance company, and the insurance business is analyzed here on the basis of its monetary components. The observation of the technical risk is thus approached through precise non-parametric estimation techniques, in order to describe both frequencies and amounts of payments. Some extensions of estimation principles and original estimators follow. Forecasting is then carried out through probabilistic modelling, moving from individual to collective models, followed by proposals for gain and risk indicators, favouring those derived from the theory of ruin. Several existing methods are then extended. These parts aim at progressively characterizing the monetary activity of an insurance company with the help of a stochastic process with jumps, from which is finally presented an original approach to cover the technical risk, illustrated by the reinsurance operation. The paper provides a particular insight into the selection of random prospects and the evaluation of risk prices.