|MONDAY 26TH||TUESDAY 27TH||WEDNESDAY 28TH||THURSDAY 29TH||FRIDAY 30TH|
|9h30 - 9h45||Welcome||9h00 - 10h00||Course 1.2||9h00 - 10h00||Course 2.2||9h00 - 10h00||Course 1.3||9h00 - 10h00||Course 1.4|
|9h45-10h45||Course 1.1||10h15-11h15||Course 2.1||10h15-11h00||Balan||10h15-11h15||Course 2.3||10h15-11h15||Drees|
|16h15-16h45||Siaulys||16h45-17h15||Bäuerle||Free Afternoon LINK||16h45-17h15||Di Bernardino||14h30-15h00||Robert|
Henrik Hult: Four lectures on importance sampling
Sophie Hautphenne: Matrix analytic methods Erlangization
|MONDAY 26TH APRIL|
15h30-16h15 Remigijus Leipus
Asymptotics of random
sums of heavy-tailed negatively dependent random variables with
Let be negatively dependent and identically distributed random variables having dominatedly varying tails, and let be a counting random variable independent of 's. In this paper, we obtain the asymptotics for the tail probability of the random sum , where the tail of is comparable with, heavier or lighter than that of .
16h15-16h45 Jonas Siaulys
Local precise large
deviation results for sums of random variables with O-regularly varying
We establish local precise large deviation results for sums of independent and identically distributed random variables with -regularly varying density and distribution function . The asymptotic behavior of the probability is comparable, for fixed , with quantities or .
16h45-17h30 Jeffrey Collamore
theory with stochastic investments and its dual financial process
This talk will be concerned with risk estimates relating to a class of random recurrence equations. Our original motivation came from the ruin problem with investments, where an insurance company invests its excess capital and earns stochastic interest on these investments. A similar problem arises when studying the stationary tail behavior of the GARCH(1,1) financial process. Both processes exhibit temporal dependence, which cannot be analyzed by classical techniques.
The tail behavior for these processes is usually obtained by observing that they satisfy a random recurrence equation, namely, where is a random function and is a random variable on . A well-known result of Goldie (1991) then states that
In this talk, we will introduce an alternative approach to Goldie's. In particular, we will begin by describing a duality connecting Cramér-Lundberg models with stochastic investments to an extended GARCH(1,1) financial process. Using this duality, we then establish the sharp upper bound
|TUESDAY 27TH APRIL|
Maume-Deschamps Multivariate risk
indicators: estimation and application to optimal reserve allocation.
We consider some risk indicators of vectorial risk processes. These indicators are expected sums of some penalties that each line of business would have to pay due to its temporary potential insolvency. The dependency between lines of business is taken into account. By using stochastic algorithms, we may estimate the minimum of these risks indicators, under a fixed total capital constraint. This minimization may apply to optimal reserve allocation.
(Joint work with Peggy Cenac and Clémentine Prieur.)
16h45-17h15 Nicole Bäuerle
Processes with Interacting Intensities
The classical models in risk theory consider a single type of claims. In the insurance business, however, several business lines with separate claim arrival processes appear naturally, and the individual claim processes may not be independent. We introduce a new class of models for such situations, where the underlying counting process is a multivariate continuous time Markov chain of pure birth type and the dependency of the components arises from the fact that the birth rate for a specific claim type may depend on the number of claims in the other component processes. Under certain conditions we obtain a fluid limit, i.e. a functional law of large numbers for these processes. We also investigate the consequences of such results for questions of interest in insurance applications. Several specific subclasses of the general model are discussed in detail and the Cramér asymptotics of the ruin probabilities are derived in particular cases.
(This is a joined work with Rudolf Grübel.)
Symbolic Computation for
Boundary Problems in Risk Theory
In this talk we will present a symbolic computation approach to boundary problems, based on operators, that is applicable in risk theory.
The main idea is to reduce the integral equations satisfied by functions of the risk processes to boundary problems. Further, one can factorize these problems into first order boundary problems, which will often allow to derive explicit expressions for the functions considered. For instance, one can find explicit expressions for the Gerber-Shiu functions in terms of the penalty function, in quite general settings.
18h00-18h30 Helene Cossette
Discrete-time risk models
based on time series for count random variables
In this talk, we consider risk models based on time series models for count random variables, which can be applied in the context of accidents. Examples of time series models for count data are integer value moving average models and integer value autoregressive models. We examine the properties of the total amount of (discounted or not) claims over a fixed number of periods. We analyze the dangerousness of the risk models through the measurement of the adjustment coefficient. Ruin measures are also examined.
Agrégation des risques
dépendants et allocation du capital
Dans cet exposé, nous considérerons un portefeuille constitué de risques dépendants (contrats d'assurance, lignes d'affaires, etc.). La relation de dépendance entre les risques est fondée sur des copules, des modèles multivariés composés ou des modèles avec mélange commun. Nous examinons diverses méthodes d'agrégation des risques dépendants adaptées aux modèles de dépendance considérés. Ces méthodes nous permettent d'aborder l'évaluation des mesures de risque Value-at-Risk (VaR) et Tail-Value-at-Risk (TVaR) et l'allocation du capital aux composantes du portefeuille de risque. Dans le cas des modèles de dépendance fondés sur les mélanges communs, nous examinons aussi la part de capital associé au risque systématique associé aux facteurs aléatoires communs induisant la dépendance entre les risques. Des exemples numériques sont présentés dans le but d'illustrer les notions présentés.
|WEDNESDAY 28TH APRIL|
10h15-11h00 Raluca Balan A
cluster limit theorem for
infinitely divisible point processes
In this talk, we examine the connection between the limit representation of an infinitely divisible point process, and its cluster representation. Our result identifies some explicit conditions for the convergence of the sequence of point processes associated to a triangular array of random variables, in terms of the probabilistic behavior of the variables in the array. As applications, we discuss the exceedance processes and the extremal index.
(Joint work with Sana Louhichi.)
11h15-12h00 John Nolan
Classes of multivariate
max stable distributions and their relationships
We examine classes of multivariate max stable distributions and look at their relationships to each other. One group of models is found by directly starting with a known angular measure. Here the class of discrete angular measures is particularly tractable in any dimension. We then explore a class of tractable (at least in two dimension) models with piecewise polynomial density for the angular measure.
Another group of models is the family of common models is the family of generalized asymmetric logistic models. We detail the connection between this class of models and the closely related class of generalized stable mixture class.
(Joint work with Anne-Laure Fougères and Cécile Mercadier)
12h00-12h30 Stilian Stoev
Tail behavior of
and limit theorems for maxima in Holder spaces
We discuss some work in progress on functional limit theorems of maxima in Holder spaces. It turns out that the classical tightness conditions of Lamperti readily apply, provided that one can control the tail-behavior of Holder norms of certain random processes. A powerful isomorphism theorem of Ciesielski allows one to obtain useful bounds on the tails of these Holder norms. Some implications on the path regularity of max-stable processes will be discussed.
|THURSDAY 29TH APRIL|
11h30-12h30 Gennady Samorodnitsky
Segments, Ruin Probabilities and the Effect of Memory on Moving Average
We obtain the rate of growth of multivariate long strange segments and the rate of decay of infinite horizon multivariate ruin probabilities for a class of infinite moving average processes with exponentially light tails. The rates are computed explicitly. We show that the rates are very similar to those of an i.i.d. process as long as moving average coefficients decay fast enough. If they do not, then the rates are significantly different. This demonstrates the change in the length of memory in a moving average process associated with certain changes in the rate of decay of the coefficients.
(Jointly with Souvik Ghosh.)
Tail empirical process
for some long memory sequences
We describes limiting behaviour of tail empirical process associated with some long memory models. We show that such process has dichotomous behaviour, according to an interplay between a Hurst parameter and a tail index. In particular, the limit may be non-Gaussian and/or degenerate, indicating an influence of long memory. On the other hand, tail empirical process with random levels never suffers from long memory. This is very desirable from a practical point of view, since such the process may be used to construct Hill estimator of the tail index. To prove our results we need to establish several new results for regularly varying distribution functions, which may be of independent interest.
(This is a joint work with Philippe Soulier.)
Elena Di Bernardino
In this work we consider the general problem of estimating the tail of a bivariate distribution. An extension of the threshold method for extreme values is developed, using a two-dimensional version of the Pickands-Balkema-de Hann Theorem. We construct a two-dimensional tail estimator and we provide its asymptotic properties. The dependence structure between the marginals is described by a copula. Simulations are implemented.
17h30-18h30 Richard Davis
Extremal Dependence for Time Series and Spatial Processes
via the Extremogram
The extremogram was developed as a tool for assessing various types of extremal dependence in a multivariate time series. The use of the extremogram in applications arising in both financial and environmental contexts will be illustrated. Currently, bootstrapping methods are being adapted to the extremogram in order to construct more meaningful and useful inference procedures. These techniques, as well as permutation procedures, will be demonstrated in several examples.
(This is joint work with Thomas Mikosch and Ivor Cribben.)
|FRIDAY 30TH APRIL|
Estimators of the Extremal Index: How Empirical Cluster Processes
Make Your Life Easy.
Recently Drees and Rootzén (2010) have introduced a very general class of empirical processes (indexed by functions) which describe certain aspects of the extreme value behavior of time series. Moreover they have proved the asymptotic normality of these processes under suitable mixing conditions.
We apply this theory to examine the asymptotic properties of smoothed blocks estimator of the extremal index. In addition, we discuss how the distribution of their estimation error can be approximated using a bootstrap approach. To this end, the limiting behavior of multiplier block bootstrap versions of the empirical processes conditional on the original data turns out to be vitally important.
Drees, H., and Rootzén, H. (2010). Limit Theorems for Empirical Processes of Cluster Functionals, to appear in the Annals of Statistics.
11h30-12h30 Jan Beirlant
Bias reduction in extreme
value methods: a personal perspective
The last decade several bias-correction methods have appeared in the estimation of the extreme value index (positive or real-valued), extreme quantiles or large return periods. One important motivation is to construct estimation methods where the results are less dependent on the choice of the threshold or the number of extremes used in the estimation. We review different methods and discuss different points of interest in this matter:
|Free afternoon on wednesday|
|List of participants|