Organizers

Cécile Mercadier and Philippe Soulier

Cécile Mercadier and Philippe Soulier

- List of participants - | |

- Group photo - |

This workshop is funded
by the ANR grant for the AST&RISK project (ANR-08-BLAN-0314-01)

SCHEDULE |

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 | |||||
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9h45-10h45 | Course 1.1 | 10h15-11h15 | Course 2.1 | 10h15-11h00 | Balan | 10h15-11h15 | Course 2.3 | 10h15-11h15 | Drees | |||||

11h15-12h15 | 11h30-12h30 | Loisel | 11h15-12h00 | Nolan | 11h30-12h30 | Samorodnitsky | 11h30-12h30 | Beirlant | ||||||

Lunch | Lunch | 12h00-12h30 | Stoev | Lunch | Lunch | |||||||||

15h30-16h15 | Leipus | 16h00-16h45 | Maume-Deschamps | Lunch | 16h00-16h45 | Kulik | 13h45-14h30 | Puccetti | ||||||

16h15-16h45 | Siaulys | 16h45-17h15 | Bäuerle | Free Afternoon LINK | 16h45-17h15 | Di Bernardino | 14h30-15h00 | Robert | ||||||

16h45-17h30 | Collamore | 17h30-18h00 | Constantinescu | 17h30-18h30 | Davis |
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18h00-18h30 | Mikosch | 18h00-18h30 | Cossette | |||||||||||

18h30-19h15 | Tang | 18h30-19h15 | Marceau |

COURSE 1 |

**Henrik Hult: Four lectures on importance sampling**

*Importance sampling in rare event simulation*

The first lecture gives an introduction to importance sampling with applications to rare event simulation. Examples include hitting probabilities of a random walk, including both the light-tailed and heavy-tailed case.*Importance sampling for computation of risk measures*

In the second lecture we consider non-linear functions of the empirical measure. Examples include computation of financial risk measures such as Value-at-Risk and Expected Shortfall.*Importance sampling and stochastic control*

There is a close connection between importance sampling and stochastic control theory. The choice of sampling distribution can be seen as the control and the objective is to minimize the relative error.*Sequential importance sampling in large state spaces*

We provide examples of importance sampling in large state spaces. Examples include the number of non-intersecting paths from to in an -by- lattice and related problems.

To Schedule |

COURSE 2 |

**Sophie Hautphenne: Matrix analytic methods
Erlangization**

- Lecture 1: Structured Markov chains and their stationary distribution

- Lecture 2: Fluid Queues

- Lecture 3: Branching processes

To Schedule |

MONDAY 26TH APRIL |

15h30-16h15 **Remigijus Leipus **
*Asymptotics of random
sums of heavy-tailed negatively dependent random variables with
applications*

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
densities*

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**
*On Cramér-Lundberg
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

as | (1) |

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

(Joint work with Anand Vidyashankar.)

18h00-18h30

We consider two different frameworks for modeling claims reserves in a non-life insurance portfolio. Due to the Poisson structure of the underlying claim counting processes one can calculate explicit expressions for the claims reserves which are understood as mean squares prediction of claim numbers and payments, as well as for the prediction error.

(Joint work with Anders H. Jessen, Muneya Matsui and Gennady Samorodnitsky.)

18h30-19h15

Consider an insurer who is exposed to a stochastic economic environment. Such an environment contains two kinds of risk. The first kind, called insurance risk, is the traditional liability risk related to the insurance portfolio and the second kind, called financial risk, is the asset risk related to the investment portfolio. Under certain regular variation conditions on the tail probabilities of the two kinds of risk, we derive some exact asymptotic formulas for the ruin probability. The formulas confirm that the ruin probability is mainly determined by the one of the two kinds of risk which is more heavy-tailed than the other.

(This talk is based on a joint work with Jinzhu Li.)

To Schedule |

TUESDAY 27TH APRIL |

11h30-12h30

We introduce longevity and mortality risks and present different models to take into account spatio-temporal dependence in stochastic mortality models. Various examples illustrate the specificity of mortality patterns in different countries and also for different sets of policyholders. Some consequences on the design of longevity derivatives are studied.

16h00-16h45 **Véronique
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**
*Multivariate Risk
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.)

17h30-18h00
**Corina Constantinescu**
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.

18h30-19h15
**Etienne
Marceau**
*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.

To Schedule |

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
Holder norms
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.

To Schedule |

THURSDAY 29TH APRIL |

11h30-12h30 **Gennady Samorodnitsky**
*Long Strange
Segments, Ruin Probabilities and the Effect of Memory on Moving Average
Processes*

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.)

16h00-16h45
**Rafal Kulik**
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.)

16h45-17h15
**Elena Di Bernardino**
*Estimating Bivariate
Tails*

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 **
Measuring
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.)

To Schedule |

FRIDAY 30TH APRIL |

10h15-11h15
**Holger Drees**
Bootstrapping Blocks
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:

- methods based on quantile models versus probability models;
- what about increase of variance;
- sensitivity with respect to the underlying second-order regular variation models;
- other tail models such as Weibull-type models;
- bias reduction in goodness-of-fit testing;
- bivariate extensions.

13h45-14h30

We propose a new algorithm to compute numerically the distribution function of the sum of d dependent, non-negative random variables with a given joint distribution.

(Joint work with P. Arbenz and P. Embrechts.)

14h30-15h00

Events with low probability but disastrous impact are of particular interest to a large variety of applied sciences. The analysis of such rare events entails the understanding of the way in which they cluster in time. Especially it is very important to measure the strength of dependence between these events. We present a new statistical methodology based on an empirical likelihood approach to estimate the distribution of the size of the clusters. Our results are illustrated through simulations and by applications to real data.

To Schedule |

Free afternoon on wednesday |

You can sleep, work, go to Marseille downtown, go to Sugiton Calanques or go to Cassis !!

How to go to Sugiton Calanques ?

From Luminy, go to Col (mountain pass) Sugiton (20 min.). Go right for a very nice view point (10 min.), you have "La Grande Candelle" (an impressive rock) at right in front of you, and the "Mont Puget" (the main summit) at left in front of you, you see the sea and small islands. You can go down to the water at Calanque Sugiton (30 min.) and take a bath, by following the trails down the valley from Col Sugiton.

From Col Sugiton you can go to the summit of Mont Puget, by following trails, and then some gravel (eboulis) climbing up a little left of the summit (60 min.). There is no danger, if you are a little careful.

How to go to Cassis ?

- By taxi (35/45 euros for 4 persons): please ask the reception TODAY to book a taxi

- By bus: Take the bus 21 at the campus exit. Alight at the stop LUMINY-VAUFREGES. Turn right at the round about and find the bus stop "NAP TOURISM". There is only one bus in the afternoon at about 1.30 pm. The ride takes about one half hour. The same bus leaves from Cassis (Gendarmerie stop) back to Luminy or Marseille 5.15, 6.30, 7 pm.

- Hiking: it takes about 6 hours to hike to Cassis from Luminy along the Calanques. It is a difficult hike. Bring lots of water.

Why go to Cassis?

- because you are afraid to get bored if you stay at the CIRM;

- because it is a nice port where you can take a scenic boat trip to the Calanques (15/20 euros).

To Schedule |

List of participants |

* Raluca Balan

* Nicole Bauerle

* Jan Beirlant

* Romain Biard

* Christophette Blanchet-Scalliet

* Margarida Brito

* Peggy Cénac-Guesdon

* Yiqing Chen

* Jeffrey Collamore

* Corina Constantinescu

* Hélène Cossette

* Bikramjit Das

* Richard Davis

* Elena Di Bernardino

* Holger Drees

* Christophe Dutang

* Anne Eyraud-Loisel

* Anne-Laure Fougères

* Armelle Guillou

* Sophie Hautphenne

* Henrik Hult

* Manel Kacem

* Rafal Kulik

* Thomas Laloe

* Claude Lefèvre

* José Rafael León

* Remigijus Leipus

* Stéphane Loisel

* Sana Louhichi

* Natalia Lysenko

* Etienne Marceau

* Esterina Masiello

* Véronique Maume-Deschamps

* Cécile Mercadier

* Thomas Mikosch

* Xavier Milhaud

* Eric Moulines

* John Nolan

* Clémentine Prieur

* Giovanni Pucetti

* Christian-Yann Robert

* Yahia Salhi

* Gennady Samorodnitsky

* Antoine Schorgen

* Jonas Siaulys

* Philippe Soulier

* Stilian Stoev

* Gilles Stupfler

* Qihe Tang

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Group photo |

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