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## Séminaires DSA

### The distribution of the supremum for spectrally asymmetric Lévy processes

Zbigniew Michna (University of Wrocław, Poland)

7 mars 2017  -  11:00-12:00, salle Extranef 118.1

In 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.

### Liste des prochaines séances

 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. 10 avril 2017 Another Look at Risk Measures in a Credibility Framework Georgios Pitselis (University of Piraeus, Greece) 14:00-15:00, salle Extranef 109 Plus d'informationHere, we introduce a new type of risk measures, the credible risk measures, in order to capture the risk of an individual  contract (or financial portfolio) as well as the industry risk consisting of several, similar but not identical, contracts (or financial portfolios). We show how risk measures can be embedded within the framework of credibility theory in the regression case. Examples are given based on the Fama/French data. 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 A family of premium principles based on mixtures of TVaRs Miguel Angel Sordo Diaz (Universidad de Cadiz, Spain) 14:00-15:00, salle Extranef 118.1 Plus d'informationRisk-adjusted distributions are commonly used in actuarial science to define premium principles. In this paper, we claim that an appropriate risk-adjusted distribution, besides of satisfying other desirable properties, should be well-behaved under conditioning with respect to the original risk distribution. Based on a sequence of such risk-adjusted distributions, we introduce a family of premium principles that gradually incorporate the degree of risk-aversion of the insurer in the safety loading. Members of this family are particular distortion premium principles than can be represented as mixtures of TVaRs, where the weights in the mixture reect the attitude toward risk of the insurer. We make a systematic study of this family of premium principles. This is joint work with A. Castaño and G. Pigueiras (University of Cádiz, Spain)
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