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March, 2013 STOCHASTIC ACTIVITY MONTH QUEUES & RISK Workshop March 4567, 2013
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. Two mini courses
will be held later in the month.
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.
MONDAY MARCH 4
TUESDAY MARCH 5
WEDNESDAY MARCH 6
THURSDAY MARCH 7
Hansjörg Albrecher Ruin and bankruptcy in risk theory In classical risk theory, the infinitetime 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 surplusdependent probability to continue despite temporary negative surplus. The consequences of this relaxation for levelcrossing 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 payout 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 riskaverse 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 riskfree 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: Andrei Badescu On some ruin problems for multidimensional risk processes
Multidimensional 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 multidimensional risk models involves an increased level of complexity
when compared to the onedimensional 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. Serban Badila 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 SparreAndersen insurance risk
model, with independence between service times (respectively claim sizes) and
interarrival times. Various quantities of interest, like the probability of
ruin, are related to performance measures of the queue, like the waiting time
distribution $W$. Yonit Barron Markovmodulated fluid flows and applications Markovmodulated 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 inventoryproduction process. Lothar Breuer Two research proposals related to Markovmodulated Brownian motion We consider two topics related to MMBMs, which are work
in progress. 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. Val Andrei Fajardo A Preemptive Accumulating Priority Queueing Model Building on the work of Stanford, Taylor, and Ziedins (2011), this talk presents a singleserver 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 LaplaceStieltjes transform of the waiting time distribution in the general multiclass 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
heavytailed 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(T1) at the prior time, and the trajectory up to it (similar
results were obtained for the Cram\'erLundberg insurance risk process). Gang Huang Limit theorems for reflected OrnsteinUhlenbeck processes We study onedimensional OrnsteinUhlenbeck 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 doublyreflected 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 samplepath 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 . 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 loglikelihood 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 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 twosided exit problem for refracted Lévy processes. SmallTime Behavior of subordinators and connection to extremal processes Based on a result in a 1987 paper by BarLev and Enis, we show that if Y_t is
a driftless subordinator with the additional property that the tail of the
Lévymeasure 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.
Bo Friis Nielsen Some multivariate exponential and gamma distributions expressed as multivariate phasetype distributions A number of multivariate exponential and gamma distributions have rational transforms. By interpreting these distributions as multivariate phasetype distributions one obtains a more streamlined presentation and the calculation of e.g. moments and crossmoments becomes routine. We present several example and highlight some research challenges still to be resolved. Zbigniew Palmowski Forwardbackward extrema of Lévy risk processes and fluid queues For a Lévy process $X$ and
fixed $S>t$ (possibly $S=+\infty$) the future updown process is defined by:
\[U^*_{t,S} = \sup_{t\leq u < t+S}(X_uX_t).\] . 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 pathdependent 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 $St$ 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 valueprocess $P_t =
P_0\exp(X_t)$ $D^*_{t,S}$ is the lowest future logreturn $\log(P_u/P_S)$ in the
timewindow $u\in[t,t+S]$, 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 heavytailed
case. When the jumps of $X$ are of single sign we explicitly identify the
onedimensional distributions in terms of the scale function. We also analyze
some examples.
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 firstpassage time problem for Levy processes and counterparty credit risk valuation For a given stochastic process X and cumulative probability distribution function H on the positive real line the inversefirst passage time problem (IFPT) is to find a function b such that the firstpassage 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 counterparty 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 quotashare 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 multidimensional 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 LaplaceStieltjes
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. Tomasz Rolski Large Mathematical problems in risk reserving In nonlife 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 Wim Schoutens Measuring systemic risk from option prices Eleni Vatamidou Corrected phasetype approximations of heavytailed risk models using perturbation analysis Numerical evaluation of ruin probabilities in heavytailed 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 phasetype and a heavytailed 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 phasetype approximation of the ruin probability. We refer to our approximations collectively as corrected phasetype approximations. For a model for which the exact ruin probability can be calculated, we check the accuracy of the corrected phasetype approximations.
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