LEFEVRE Claude

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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.

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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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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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.

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).

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).

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.

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.

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.

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

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