GOFFARD Pierre Olivier

The resource-consuming mining of blocks on a blockchain equipped with a proof of work consensus protocol bears the risk of ruin, namely when the operational costs for the mining exceed the received rewards. In this paper we investigate to what extent it is of interest to join a mining pool that reduces the variance of the return of a miner for a specified cost for participation. Using methodology from ruin theory and risk sharing in insurance, we quantitatively study the effects of pooling in this context and derive several explicit formulas for quantities of interest. The results are illustrated in numerical examples for parameters of practical relevance.

Insurance loss distributions are characterized by a high frequency of small amounts and a lower, but not insignificant, occurrence of large claim amounts. Composite models, which link two probability distributions, one for the "belly" and the other for the "tail" of the loss distribution, have emerged in the actuarial literature to take this specificity into account. The parameters of these models summarize the distribution of the losses. One of them corresponds to the breaking point between small and large claim amounts. The composite models are usually fitted using maximum likelihood estimation. A Bayesian approach is considered in this work. Sequential Monte Carlo samplers are used to sample from the posterior distribution and compute the posterior model evidences to both fit and compare the competing models. The method is validated via a simulation study and illustrated on insurance loss datasets.

The probability of successfully spending twice the same bitcoins is considered. A double-spending attack consists in issuing two transactions transferring the same bitcoins. The first transaction, from the fraudster to a merchant, is included in a block of the public chain. The second transaction, from the fraudster to himself, is recorded in a block that integrates a private chain, exact copy of the public chain up to substituting the fraudster-to-merchant transaction by the fraudster-to-fraudster transaction. The double-spending hack is completed once the private chain reaches the length of the public chain, in which case it replaces it. The growth of both chains are modeled by two independent counting processes. The probability distribution of the time at which the malicious chain catches up with the honest chain, or equivalently the time at which the two counting processes meet each other, is studied. The merchant is supposed to await the discovery of a given number of blocks after the one containing the transaction before delivering the goods. This grants a head start to the honest chain in the race against the dishonest chain.

Approximations for an unknown density g in terms of a reference density fν and its associated orthonormal polynomials are discussed. The main application is the approximation of the density f of a sum S of lognormals which may have different variances or be dependent. In this setting, g may be f itself or a transformed density, in particular that of log S or an exponentially tilted density. Choices of reference densities fν that are considered include normal, gamma and lognormal densities. For the lognormal case, the orthonormal polynomials are found in closed form and it is shown that they are not dense in L2(fν), a result that is closely related to the lognormal distribution not being determined by its moments and provides a warning to the most obvious choice of taking fν as lognormal. Numerical examples are presented and comparison are made to established approaches such as the Fenton–Wilkinson method and skew-normal approximations. Also extension to density estimation for statistical data sets and non-Gaussian copulas are outlined.

Goodness-of-fit procedures are provided to test the validity of compound models for the total claims, involving specific laws for the constituent components, namely the claim frequency distribution and the distribution of individual claim sizes. This is done without the need for observations on these two component variables. Goodness-of-fit tests that utilize the Laplace transform as well as classical tools based on the distribution function, are proposed and compared. These methods are validated by simulations and then applied to insurance data. MSC 2010: 60G55, 60G40, 12E10.

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.

The purpose of this thesis is to study numerical methods for approximating the probability density associated with random variables that have compound distributions. These random variables are commonly used in actuarial science to model the risk borne by a portfolio of contracts. In catastrophe theory, the probability of ultimate catastrophe in the compound Poisson model is equal to the survival function of a compound geometric distribution. The proposed numerical method consists of an orthogonal projection of the density onto a basis of orthogonal polynomials. These polynomials are orthogonal with respect to a reference probability measure belonging to the Quadratic Natural Exponential Families. The polynomial approximation method is compared to other density approximation methods based on moments and the Laplace transform of the distribution. The extension of the method in dimension higher than $1$ is presented, as well as the obtention of a density estimator from the approximation formula. This thesis also includes the description of an aggregation method adapted to portfolios of life insurance contracts of individual savings type. The aggregation procedure leads to the construction of model points to allow the evaluation of the best estimate reserves in a reasonable time and in accordance with the European Solvency II directive.

An aggregation method adapted to life insurance portfolios is presented. We aim at optimizing the computation time when using Monte Carlo simulations for best estimate liability calculation. The method is a two-step procedure. The first step consists in using statistical partitioning methods in order to gather insurance policies. The second step is the construction of a representative policy for each aforementioned groups. The efficiency of the aggregation method is illustrated on a real saving contracts portfolio within the frame of a cash flow projection model used for best estimate liabilities and solvency capital requirements computations. The procedure is already part of AXA France valuation process.