# European Institute for Statistics, Probability, Stochastic Operations Research and their Applications

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March, 2013

STOCHASTIC ACTIVITY MONTH

QUEUES & RISK

Workshop

March 4-5-6-7,  2013

 SUMMARY REGISTRATION SPEAKERS PROGRAMME ABSTRACTS

Both queueing and risk theory are active research areas in applied probability. Although queueing and risk models are often, via a common underlying random walk and a duality argument, related to each other, the two fields developed to a large extent independent of each other. Partly this is because their application areas are different and partly because performance measures of interest in the two fields are usually different.

The workshop will consist of both longer talks by more senior participants, and shorter talks by more junior participants. The workshop will be structured so as to stimulate interaction between the participants. In particular, ample lunch breaks, coffee breaks and a conference dinner will be scheduled.

Two mini courses will be held later in the month.

 Serban Badila TU Eindhoven Onno Boxma TU Eindhoven Martijn  Pistorius University of Amsterdam Jacques Resing TU Eindhoven

CONFIRMED SPEAKERS

 Hansjörg Albrecher Université de Lausanne Søren Asmussen Aarhus University Florin Avram Université de Pau Andrei Badescu University of Toronto Serban Badila TU Eindhoven Yonit Barron University of Haifa Lothar Breuer University of Kent Bernardo D'Auria Universidad Carlos III de Madrid Val Andrei Fajardo University of Waterloo Sergey Foss Heriot-Watt University (Edinburgh) and Institute of Mathematics, Novosibirsk Gang Huang University of Amsterdam Jevgenijs Ivanovs Université de Lausanne Dominik Kortschak Joanneum Research Ronnie Loeffen University of Manchester Andreas Löpker Helmut Schmidt University Bo Friis Nielsen Technical University of Denmark Zbigniew Palmowski University of Wroclaw David Perry University of Haifa Martijn Pistorius University of Amsterdam Landy Rabehasaina Université de Franche-Comté Jacques Resing TU Eindhoven Tomasz Rolski University of Wroclaw Wim Schoutens KU Leuven Eleni Vatamidou TU Eindhoven

REGISTRATION

There is no deadline for registration, but the number of places is limited. For organizational reasons registration is obligatory for all participants (organizers and speakers included).

Please indicate on the registration form your attendance, participation in the lunches, dinner.

For invited speakers hotel accommodation will be arranged by the organization. You are requested to indicate the arrival and departure dates on the registration form.

All other participants have to arrange their own hotel booking.
We have a "preferred" hotel, which can be booked at a special rate, please email Patty Koorn for instructions on how to make use of this special offer.
Or you may try other hotels around Eindhoven University of Technology (please note: prices listed are "best available").
There more hotel options on
the webpages of the Tourist Information Eindhoven, Postbus 7, 5600 AA Eindhoven.

PROGRAMME

MONDAY MARCH 4

 10.00 - 10.05 Opening                                     by Remco van der Hofstad (scientific director Eurandom) 10.05 - 10.50 Hans-Jörg Albrecher Ruin and bankruptcy in risk theory 10.50 - 11.35 Tomasz Rolski Large Mathematical problems in risk reserving 11.35 - 12.00 Break 12.00 - 12.45 Wim Schoutens Measuring systemic risk from option prices 12.45 - 14.00 Lunch 14.00 - 14.30 Bernardo D'Auria Deciding if to join or to balk a queueing network by having only partial information about it 14.30 - 15.00 Val Andrei Fajardo A Preemptive Accumulating Priority Queueing Model 15.00 - 15.30 Break 15.30 - 16.15 Lothar Breuer Two research proposals related to Markov-modulated Brownian motion

TUESDAY MARCH 5

 10.00 - 10.45 Andrei Badescu On some ruin problems for multi-dimensional risk processes 10.45 - 11.30 Landy Rabehasaina Risk processes in dimension $2$ 11.30 - 12.00 Break 12.00 - 12.45 Jacques Resing Queues and risk models with simultaneous arrivals 12.45 - 14.00 Lunch 14.00 - 14.30 Andreas Löpker Small-Time Behavior of subordinators and connection to extremal processes 14.30 - 15.00 Dominik Kortschak Ruin problems for processes in a changing environment 15.00 - 15.30 Break 15.30 - 16.15 Martijn Pistorius On an inverse first-passage time problem for Levy processes and counter-party credit risk valuation

WEDNESDAY MARCH 6

 10.00 - 10.45 Sergey Foss On exceedance times for some processes with dependent increments 10.45 - 11.30 Ronnie Loeffen Overshoots of spectrally negative Lévy processes and applications 11.30 - 12.00 Break 12.00 - 12.45 Jevgenijs Ivanovs Sequential testing for Erlang distributions and scale matrices 12.45 - 14.00 Lunch 14.00 - 14.30 Yonit Barron Markov-modulated fluid flows and applications 14.30 - 15.00 Gang Huang Limit theorems for reflected Ornstein-Uhlenbeck processes 15.00 - 15.30 Break 15.30 - 16.15 David Perry Perishable Inventory Systems with Random Replenishments 18.30 - Conference dinner

THURSDAY MARCH 7

 10.00 - 10.45 Florin Avram On positive approximations by sums of exponentials and by matrix exponential functions 10.45 - 11.30 Bo Friis Nielsen Some multivariate exponential and gamma distributions expressed as multivariate phase-type distributions 11.30 - 12.00 Break 12.00 - 12.45 Zbigniew Palmowski Forward-backward extrema of Lévy risk processes and fluid queues 12.45 - 14.00 Lunch 14.00 - 14.30 Eleni Vatamidou Corrected phase-type approximations of heavy-tailed risk models using perturbation analysis 14.30 - 15.00 Serban Badila Dependencies in risk models and their dual queueing models 15.00 - 15.30 Break 15.30 - 16.15 Søren Asmussen Porfolio size as function of the premium: modeling and optimization

ABSTRACTS

Hansjörg Albrecher

Ruin and bankruptcy in risk theory

In classical risk theory, the infinite-time ruin probability of a surplus process is calculated as the probability of the process becoming negative at some point in time. In this talk a relaxation of the ruin concept to the concept of bankruptcy is discussed, according to which one has a positive surplus-dependent probability to continue despite temporary negative surplus. The consequences of this relaxation for level-crossing probabilities, related quantities, optimal dividend strategies as well as interpretations in a related queueing model are studied.

Søren Asmussen

Porfolio size as function of the premium: modeling and optimization

How to set the premium size is obviously one of the main decisions to be made by an insurance company, but quantitative studies of the effect of lowering or increasing the premium are few in the literature compared to other control strategies, say for dividend pay-out or reinsurance. Intuitively one expects that lowering the premium p per customer will increase the portfolio size n(p)(the number of people insured) but it is less clear what is the effect on the net income pn(p), on the gain, the ruin probability etc. One difficulty is to quantify the dependence of n(p) on p and of the phenomenon of adverse selection (a higher premium will bias the portfolio towards customers prone to many claims and thereby less attractive to the company). We formulate a general criterion for a risk-averse customer to insure, based on calculations of his present values of the alternative strategies of insuring or not insuring and further parameters such as his discount rate d and the risk-free interest rate r<d. Implications for the classical empirical Bayes model with the rate of claims of a customer being a random variable are derived and extensions given to situations with customers having only partial information on their rate or/and stochastic discount rates. As example of a control problem, minimizing the ruin probability as function of p is studied. Joint work with Bent Jesper Christensen and Michael Taksar.

Florin Avram

On positive approximations by sums of exponentials and by matrix exponential functions

We review some problems concerning moments based approximations of densities,  including:
1)  The characterization  of the domain of nonnegativity of sums  of exponentials with given real exponents.
2) The analog problem for polynomials, or mixed  Erlang distributions.
3) The unification of these problems under the moments parametrization.
4)  The "completion problem", consisting in the  determination of the minimal bidiagonal/triangular order of positive realizations in the case  the original Laplace transform has only real poles, and the determination of the poles which must be added.
(joint work with B. Dumitrescu)

On some ruin problems for multi-dimensional risk processes

Multi-dimensional risk theory represents a topic that gained popularity in recent years and has potential connections with models encountered in queueing theory. The analysis of the class of multi-dimensional risk models involves an increased level of complexity when compared to the one-dimensional case. This happens mainly due to the various dependence structures assumed among the surplus processes under consideration. The present talk focuses on some of the challenges that are encountered in the analysis of ruin related risk measures for two particular scenarios.
The first model considers a proportional reinsurance case and represents an extension of a model proposed by Avram et al. (2008). The first insurer is subject to claims arising from two independent compound Poisson processes. The second insurer that can be viewed as a different line of business of the same insurer, or as a reinsurer, covers a proportion of the claims caused by one of these two compound Poisson processes. Simple probabilistic and geometric arguments lead to the calculation of the Laplace transform of the ruin time, when ruin is defined as the first exit from the positive quadrant.
The second model considers a more general scenario with claim arrivals introduced via common shocks. A multivariate Gerber-Shiu discounted penalty function is introduced and analyzed in a recursive way under various claim assumptions. Numerical examples and future research ideas are discussed in the end.

Dependencies in risk models and their dual queueing models

It is well known that there are duality relations between the classical $GI/GI/1$ queue and the corresponding classical Sparre-Andersen insurance risk model, with independence between service times (respectively claim sizes) and inter-arrival times. Various quantities of interest, like the probability of ruin, are related to performance measures of the queue, like the waiting time distribution $W$.
Under stability conditions, when starting the risk process at an arrival epoch and with initial capital $u$, the infinite horizon ruin probability, $\psi_0(u)$ is equal to the tail $\mathbb P(W>u)$ of the steady state workload in the dual queue. This duality is a path-wise identity and no independence assumptions are needed.
We consider models that exhibit some kind of dependence structure. For example the size of the current claim may depend on its inter-arrival time in such a way that the above performance measures are tractable. In the dual queueing setting this means the current service requirement and the subsequent inter-arrival time are correlated.
The dependence structure to be presented is modeled by a class of bivariate matrix-exponential distributions (as in Bladt \& Nielsen 2010) in which the joint Laplace-Stieltjes transform of the claim size and the inter-claim time is a rational function. This class generalizes some bivariate Gamma models already considered in the literature.
We obtain exact results for the infinite-horizon ruin probabilities and for the waiting time in the dual queueing model. We also derive a relation between the workload and waiting time distributions, and we give the connection with the ruin probability in the delayed risk model.

Yonit Barron

Markov-modulated fluid flows and applications

Markov-modulated fluid flow models have been an active area of research in recent years. Several applications of the fluid flows will be introduced related to the insurance and the inventory-production process.
In particularly, we consider a make-to-stock inventory model with double reflected finite switchovers; one positive and one negative. The production process alternates between two pre-determined production rates - a positive drift and a negative drift. The system is supposed to have a finite storage capacity. If the system is full the production is stopped; the production restarts after the next customer arrival. Any demand which cannot be fulfilled immediately is backlogged up to a certain level. Below this level it is lost. Four types of costs are imposed: holding cost, production loss costs, backlog costs and unsatisfied demand costs. By applying the optional sampling theorem to the Asmussen & Kella martingale and by using the fluid flow application by Ramaswami we obtain the discounted cost functionals.

Lothar Breuer

Two research proposals related to Markov-modulated Brownian motion

We consider two topics related to MMBMs, which are work in progress.
First we look at MMBMs with jumps from a boundary. Starting at a positive value, the process under study evolves like an MMBM until it hits the value zero. At this moment it jumps upwards according to a given distribution function, after which the MMBM movement continues until the level zero is hit again, followed by another jump, and so on.
For this process we determine the stationary distribution via Markov renewal theory. This topic allows several extensions and variants for further research.
A second topic is the relationship between a class of Markovian binary trees (MBTs) and their exploration excursions, which are fluid flows.
The tree profile may be determined as the family of local times of the related fluid flow. Further research may consider the limits of the renormalised MBTs and their respective exploration excursions, leading to an extension of the second Ray-Knight theorem.

Bernardo D'Auria

Deciding if to join or to balk a queueing network by having only partial information about it

Customers perceive the reception of a service as a reward and at the same time they consider the time spent waiting as a cost. Assuming that they can take an action on arriving, that is, to join or to balk the system, it is interesting to ask if it is possible that an equilibrium   strategy may exist. Of course, customers' decisions are based on the information they get about the current state of the system. More information they get, closer will be their prediction for the waiting cost.
In this talk we analyze the case when the system is made of a queueing network, in specific a tandem network with two servers. Customers will receive only partial information about the state of the system and we show how to compute the individual threshold optimal strategy. To different strategies it will correspond different individual expected rewards, as well as different expected network throughput.

Val Andrei Fajardo

A Preemptive Accumulating Priority Queueing Model

Building on the work of Stanford, Taylor, and Ziedins (2011), this talk presents a single-server accumulating preemptive priority queueing model in which waiting customers accumulate priority credit at a linear rate that depends upon their priority classification. Fundamental elements of the service time structure are generalized and expanded in order to obtain the Laplace-Stieltjes transform of the waiting time distribution in the general multi-class accumulating priority queue.

Sergey Foss

On exceedance times for some processes with dependent increments

Consider a random walk Z(n) with a negative drift and i.i.d. increments with heavy-tailed distribution, and let M be its supremum. Asmussen and Klüppelberg (1996) studied the behavior of the random walk given that M>x, for x large, and obtained a limit theorem, as x increases to infinity, for the distribution of the quadruple that includes time T to exceed level x, position Z(T) at this time, position Z(T-1) at the prior time, and the trajectory up to it (similar results were obtained for the Cram\'er-Lundberg insurance risk process).
We propose another proof of the result of AK (1996) and formulate several extensions to various regenerative-type models and, in particular, to the case of a random walk with dependent increments. We also consider a number of related problems.
(joint work with S Asmussen)

Gang Huang

Limit theorems for reflected Ornstein-Uhlenbeck processes

We study one-dimensional Ornstein-Uhlenbeck processes, with the distinguishing feature that they are reflected on a single boundary (put at level 0) or two boundaries (put at levels 0 and d > 0). In the literature they are referred to as reflected OU (ROU ) and doubly-reflected OU (DROU ) respectively. For both cases, we explicitly determine the decay rates of the (transient) probability to reach a given extreme level. The methodology relies on sample-path large deviations, so that we also identify the associated most likely paths. For DROU , we also consider the loss process Ut , that is, the local time at upper boundary d. We derive a central limit theorem (C LT ) for Ut , using techniques from stochastic integration and the martingale C LT .
(joint work with Michel Mandjes. Peter Spreij)

Jevgenijs Ivanovs

Sequential testing for Erlang distributions and scale matrices

We consider sequential probability ratio test for two simple hypotheses in the class of Erlang(k) distributions (or functionals of those). The corresponding log-likelihood ratio under both hypotheses evolves as a random walk with increment cY − d, where Y has Erlang(k, λ) distribution and λ is either λ0 or λ1. This test is stopped and a decision is made when the likelihood ratio exits a certain interval, boundaries of which  are to be chosen in an optimal way. Hence analysis of our test reduces to the analysis of the two- sided exit problem for certain random walks. We accomplish the latter using scale matrices associated to Markov additive processes which embed the random walks of interest. It turns out that the scale matrices corresponding to our two hypotheses satisfy a very simple relation. Furthermore, we find an explicit representation
of the scale matrices, which then leads to explicit equations for the decision boundaries. In addition, we determine the expected number of observations under both hypotheses. This work extends [1] and puts it into modern context. It also provides an interesting link between statistics and ruin theory.

Dominik Kortschak

Ruin problems for processes in a changing environment

In this talk we consider risk processes in a time changing environment. Our main motivation comes from climate change. The principal idea is that in the future there will be more claims that will be more severe. We study some specific models that have  this behavior (with heavy tailed as well as for light tailed claim sizes) and provide asymptotic results for the ruin probability. Further we provide a result on the insurability of risks with infinite mean.

Ronnie Loeffen

Overshoots of spectrally negative Lévy processes and applications

For a Lévy process with no upward jumps, we are interested in the distribution of the overshoot, which is the value of the process at the time it first drops below zero. In particular, we discuss a technique which gives analytical expressions for this distribution and for the slightly more general, expected discounted penalty function. In order to illustrate the usefulness of these expressions, we consider two applications, namely occupation times for spectrally negative Lévy processes and the two-sided exit problem for refracted Lévy processes.

Andreas Löpker

Small-Time Behavior of subordinators and connection to extremal processes

Based on a result in a 1987 paper by Bar-Lev and Enis, we show that if Y_t is a driftless subordinator with the additional property that the tail of the Lévy-measure behaves like -c*log(x) as x tends to zero, then -t*log(Y_t) tends weakly to a limit having an exponential distribution. We investigate several equivalent conditions to ensure this convergence and present examples of processes that fulfill these conditions. We then prove that one can extend these results to a statement about convergence of processes and show that under the above conditions -t*log(Y_{ts}) tends weakly to what is called an extremal process. Moreover, we present more general results concerning the convergence.
(joint work with Shaul Bar-Lev, Offer Kella and Wolfgang Stadje)

Bo Friis Nielsen

Some multivariate exponential and gamma distributions expressed as multivariate phase-type distributions

A number of multivariate exponential and gamma distributions have rational transforms. By interpreting these distributions as multivariate phase-type distributions one obtains a more streamlined presentation and the calculation of e.g. moments and cross-moments becomes routine. We present several example and highlight some research challenges still to be resolved.

Zbigniew Palmowski

Forward-backward extrema of Lévy risk processes and fluid queues

For a Lévy process $X$ and fixed $S>t$ (possibly $S=+\infty$) the future up-down process is defined by: $U^*_{t,S} = \sup_{t\leq u < t+S}(X_u-X_t).$ . The fluctuations of $U^*_{t,S}$ are described by the running supremum and running infimum: $\overline{U}^*_{T,S} = \sup_{0\leq t\leq T} U^*_{t,S}, \qquad\underline{U}^*_{T,S} = \inf_{0\leq t\leq T} U^*_{t,S}.$ .The random variables $\overline U^*_{T,S}$ and $\underline U^*_{T,S}$ are path-dependent performance measures. For a fluid queue $U^*_{t,S}$ describes the buffer content of fluid queues observed at time $t$ when queue already has been already running $S-t$ units of time before $0$ and $\overline{U}^*_{T,S}$ and $\underline{U}^*_{T,S}$ are the maximal and minimal such contents for $t$ ranging over $[0,T]$. Similarly, for a financial asset with value-process $P_t = P_0\exp(X_t)$ $D^*_{t,S}$ is the lowest future log-return $\log(P_u/P_S)$ in the time-window $u\in[t,t+S]$, and $\overline D^*_{T,S}$ and $\underline D^*_{T,S}$ are the maximal and minimal such future returns for $t$ ranging over $[0,T]$. In the case that $X$ has strictly negative mean we find the exact asymptotic decay of the tail distributions of above extrema in both the Cram\'er and heavy-tailed case. When the jumps of $X$ are of single sign we explicitly identify the one-dimensional distributions in terms of the scale function. We also analyze some examples.
(joint work with E. Baurdoux and M. Pistorius)

David Perry

Perishable Inventory Systems with Random Replenishments

A guide to perishable inventory systems (PIS's) that are refilled by randomly arriving items and not by ordering decisions is introduced. The literature on this class of PIS's (for which a blood bank or an organ trans- plantation center are prominent examples) is sparse. The survey starts with the pioneering work on a prototype model in which item arrivals and demand arrivals form independent Poisson processes. We show how to compute all per- formance measures of interest for this PIS. Thereafter, extensions in several directions are reviewed, among them (i) PIS's with finite capacity and waiting demands; (ii) PIS's with renewal item arrival times; (iii) batch arrivals of items or demands; (iv) actuarial valuation; (v) optimization and control. Some novel contributions are also introduced.

Martin Pistorius

On an inverse first-passage time problem for Levy processes and counter-party credit risk valuation

For a given stochastic process X and cumulative probability distribution function H on the positive real line the inverse-first passage time problem (IFPT) is to find a function b such that the first-passage time of X below b is distributed according to H. In this talk we consider the IFPT for a Levy process, and discuss applications to the valuation of financial contracts that are subject to counter-party credit risk.

Landy Rabehasaina

Risk processes in dimension $2$

We present two situations where one can obtain information on the (properly defined) ruin probability of a bivariate risk process, of which entries represent two business lines of an insurance company, or an insurance and reinsurance company that share common claims. One features an interest force, with claims incoming according to a Poisson process and general claims distribution. The other one features one type of claims incoming from a (possibly modulated) Poisson process with general and light tailed claims, as well as another one modelled thanks to a fractional brownian motion. Both share their claims according to a quota-share policy. In the first specific case, a simple geometric argument yields the cdf of the ruin time. In the second case, asymptotics are provided when initial reserves tend to infinity along a fixed direction.

Jacques Resing

Queues and risk models with simultaneous arrivals

We focus on a particular connection between queueing and risk models in a multi-dimensional setting. We first consider the joint workload process in a queueing model with parallel queues and simultaneous arrivals at the queues. For the case that the service times are ordered (from largest in the first queue to smallest in the last queue) we obtain the Laplace-Stieltjes transform of the joint stationary workload distribution. Using a multivariate duality argument between queueing and risk models, this also gives the Laplace transform of the survival probability of all books in a multivariate risk model with simultaneous claim arrivals and the same ordering between claim sizes.
(joint work with Onno Boxma, Serban Badila and Erik Winands)

Tomasz Rolski

Large Mathematical problems in risk reserving

In non-life actuarial practice insurers are obliged to make reserves against future claims. Different methods are used, however hardly any has a stochastic model in the background. In few papers recently probabilistic models of reserves are built and next estimators for future reserves are computed. In this talk we want to propose a more realistic model, which is however more difficult in analysis. On the
macroscopic side we consider $N(1)$ claims $T_1,\ldots,T_{N(1)}$ (unordered sequence) arriving in $[0,1]$ accroding to a Poisson point process, possibly non-homogeneous and the distribution of the total payment of each claim is a Poisson compound. On the microscopic side we assume that each claim initiates a stream of iid payments $C$, which form non-homogeneous Poisson processes of payment moments.
Different problems leads to study expressions of the form $${\bf E}[\Phi(N(1),T_1,\ldots,T_{N(1)}) | \sum_{j=1}^ZC_j=k],$$ where $Z$ is a mixed Poisson random variable with mean depending on a
symmetric function of $T_1,\ldots,T_{N(1)}$ and $\Phi(n,t_1,\ldots,t_n)$ is for each $n$ a symmetric.
In the talk we will discuss a saddle point approximations for such functionals.
(joint work with Agata Tomanek)

Wim Schoutens

Measuring systemic risk from option prices

Eleni Vatamidou

Corrected phase-type approximations of heavy-tailed risk models using perturbation analysis

Numerical evaluation of ruin probabilities in heavy-tailed risk models is an important and challenging problem. We present very accurate approximations of the ruin probability that capture the tail behavior of the exact ruin probability and provide a small relative error. Motivated by statistical analysis, we assume that the claim sizes are a mixture of a phase-type and a heavy-tailed distribution, and with the aid of perturbation analysis we derive a series expansion for the ruin probability. Our proposed approximations consist of the first two terms of this series expansion, where the first term is a phase-type approximation of the ruin probability. We refer to our approximations collectively as corrected phase-type approximations. For a model for which the exact ruin probability can be calculated, we check the accuracy of the corrected phase-type approximations.

Practical information

Conference Location
The workshop location is Eurandom,  Den Dolech 2, 5612 AZ Eindhoven, METAFORUM, 4th floor, MF 12.

EURANDOM is located on the campus of Eindhoven University of Technology, in the new METAFORUM building (no. 44 on the map). The university is located at 10 minutes walking distance from Eindhoven railway station (take the exit north side and walk towards the tall building on the right with the sign TU/e).

For all information on how to come to Eindhoven, please check http://www.eurandom.tue.nl/contact.htm

Contact
Coordinator Eurandom

#### Last updated 25-04-13, by PK

P.O. Box 513, 5600 MB Eindhoven, The Netherlands
tel. +31 40 2478100
e-mail: info@eurandom.tue.nl