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## Séminaires DSA

### Discretization error for the maximum of a Gaussian field

Jean-Marc Azaïs (Université de Toulouse, France)

4 avril 2017  -  14:00-15:00, salle Extranef 109

A Gaussian field $X$ defined on a square $T$ of $\R^2$ is considered. We assume that this field is only observed at some points of a regular grid with spacing $\frac{1}{n}$. We are interested in the discretization error $M - M_n$, with $M$ the global maximum of $X$ over $T$ and $M_n$ the maximum of $X$ over the observation grid. Using a model inspired by Slepian models, an asymptotic equivalent of the discretization error is given and thus an asymptotic bound for this error.

### Liste des prochaines séances

 aujourd'hui Optimal barriers in a modified surplus process Başak Bulut Karageyik (Hacettepe University, Ankara, Turkey) 11:00-12:00, salle Extranef 125 Plus d'informationWe obtain the optimal pair of initial surplus and barrier level in a lower barrier model of a modified surplus process. In particular, we examine the defective distribution function of the time to ruin with given lower barrier and initial surplus which is suggested by Nie et al. [Minimizing the ruin probability through capital injections. Ann Actuar Sci. 2011;5(2):195–209]. We aim to take this approach one step further by proposing optimal reinsurance under the minimum finite time ruin probability and maximum benefit criteria such as the released capital, expected profit and expected utility. We calculate the optimal pairs of initial surplus and barrier levels for different time periods, loading factors and weights of the criteria. We analyse the robustness of the results by a sensitivity analysis. 7 mars 2017 The distribution of the supremum for spectrally asymmetric Lévy processes Zbigniew Michna (University of Wrocław, Poland) 11:00-12:00, salle Extranef 118.1 Plus d'informationIn this article we derive formulas for the probability $P(\sup_{t\leq T} X(t)>u)$, $T>0$ and $P(\sup_{t<\infty} X(t)>u)$ where $X$ is a spectrally positive Lévy process with infinite variation. The formulas are generalizations of the well-known Takács formulas for stochastic processes with non-negative and interchangeable increments. Moreover, we find the joint distribution of $\inf_{t\leq T} Y(t)$ and $Y(T)$ where $Y$ is a spectrally negative Lévy process.  Joint work with Zbigniew Palmowski, Martijn Pistorius. 4 avril 2017 Discretization error for the maximum of a Gaussian field Jean-Marc Azaïs (Université de Toulouse, France) 14:00-15:00, salle Extranef 109 Plus d'informationA Gaussian field $X$ defined on a square $T$ of $\R^2$ is considered. We assume that this field is only observed at some points of a regular grid with spacing $\frac{1}{n}$. We are interested in the discretization error $M - M_n$, with $M$ the global maximum of $X$ over $T$ and $M_n$ the maximum of $X$ over the observation grid. Using a model inspired by Slepian models, an asymptotic equivalent of the discretization error is given and thus an asymptotic bound for this error. 5 mai 2017 Ruin probabilities in risk models with dependent and phase–type distributed claims and inter-arrivals Mogens Bladt (National University of Mexico and University of Copenhagen, Denmark) 14:00-15:00, salle Extranef 118.1 Plus d'informationIn this talk we consider risk-reserve processes where we allow for dependency between claims and inter--arrivals in several ways. Between claims we may either have a (deterministic) linear increase in the reserve or a stochastic development governed by a Brownian motion with a drift. Assuming claims and inter--arrivals being phase--type distributed, we develop methods for calculating explicit or exact ruin probabilities of different kind (classical infinite horizon, Parisian and finite--time Parisian) by representing the original risk--reserve process in terms of an equivalent fluid flow model with an optional Brownian component. We shall pay special attention to the construction and control of the (Pearson) correlation between claim sizes and inter-arrival times using a copula method based on order statistics for the construction of bivariate phase—type distributions. We provide a numerical study regarding the effect of the correlation and different scenarios. 12 mai 2017 Miguel Angel Sordo Diaz (Universidad de Cadiz, Spain) 14:00-15:00, salle Extranef 118.1
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