Faults Self-Organized by Repeated Earthquakes in a Quasi-Static Antiplane Crack Model

We study a 2D quasi-static discrete {\it crack} anti-plane model of a tectonic plate with long range elastic forces and quenched disorder. The plate is driven at its border and the load is transfered to all elements through elastic forces. This model can be considered as belonging to the class of self-organized models which may exhibit spontaneous criticality, with four additional ingredients compared to sandpile models, namely quenched disorder, boundary driving, long range forces and fast time crack rules. In this ''crack'' model, as in the ''dislocation'' version previously studied, we find that the occurrence of repeated earthquakes organizes the activity on well-defined fault-like structures. In contrast with the ''dislocation'' model, after a transient, the time evolution becomes periodic with run-aways ending each cycle. This stems from the ''crack'' stress transfer rule preventing criticality to organize in favor of cyclic behavior. For sufficiently large disorder and weak stress drop, these large events are preceded by a complex space-time history of foreshock activity, characterized by a Gutenberg-Richter power law distribution with universal exponent $B=1 \pm 0.05$. This is similar to a power law distribution of small nucleating droplets before the nucleation of the macroscopic phase in a first-order phase transition. For large disorder and large stress drop, and for certain specific initial disorder configurations, the stress field becomes frustrated in fast time : out-of-plane deformations (thrust and normal faulting) and/or a genuine dynamics must be introduced to resolve this frustration

Convergence and asymptotic variance of bootstrapped finite-time ruin probabilities with partly shifted risk processes.

The classical risk model is considered and a sensitivity analysis of finite-time ruin probabilities is carried out. We prove the weak convergence of a sequence of empirical finite-time ruin probabilities. So-called partly shifted risk processes are introduced, and used to derive an explicit expression of the asymptotic variance of the considered estimator. This provides a clear representation of the influence function associated with finite time ruin probabilities, giving a useful tool to quantify estimation risk according to new regulations.Finite-time ruin probability; robustness; Solvency II; reliable ruin probability; asymptotic normality; influence function; partly shifted risk process; Estimation Risk Solvency Margin. (ERSM).

Robustness analysis and convergence of empirical finite-time ruin probabilities and estimation risk solvency margin.

We consider the classical risk model and carry out a sensitivity and robustness analysis of finite-time ruin probabilities. We provide algorithms to compute the related influence functions. We also prove the weak convergence of a sequence of empirical finite-time ruin probabilities starting from zero initial reserve toward a Gaussian random variable. We define the concepts of reliable finite-time ruin probability as a Value-at-Risk of the estimator of the finite-time ruin probability. To control this robust risk measure, an additional initial reserve is needed and called Estimation Risk Solvency Margin (ERSM). We apply our results to show how portfolio experience could be rewarded by cut-offs in solvency capital requirements. An application to catastrophe contamination and numerical examples are also developed.Finite-time ruin probability; robustness; Solvency II; reliable ruin probability; asymptotic Normality; influence function; Estimation Risk Solvency Margin (ERSM)