It is known how one can define a measure under which a Levy
process starts
at x>0 and stays positive. It is also known how to do this when x=0,
although this
requires a different construction. What was not known, in general, was
whether the first
measure conveges to the second as x decreases to 0. I will show that
this is actually the
case (joint work with Loic Chaumont, Univ Paris vi) and then use this
fact to give a
different proof of some recent results of Kyprianou et al on the 2-sided
exit problem for
the reflected process derived from a spectrally one-sided Levy process.
I plan to discuss various notions of coupling for Markov chains and
stochastically recursive sequences (``ordinary'' coupling, strong
coupling, backwards coupling, coupling-from-the past) and their
relation to perfect simulation.
A functional approach is taken for the total claim amount distribution
for the individual risk model. Various commonly used approximations
for this distribution are considered, including the compound Poisson
approximation, the compound binomial approximation, the compound negative
binomial approximation and the normal approximation. New
approximation formulae are obtained as refinements to the existing
approximations. Other applications of the functional approach to
quantities of interest in risk theory are discussed.
Epidemics are spatio-temporal processes by definition, and yet due to
the difficulty in collecting data on the spatial distribution of hosts,
epidemics data often ignore space or treat it in an overly-simplistic
fashion. In this talk I present data collected in Cambridge on the
spread of a fungal pathogen through crops, in which the spatial
distribution of hosts is recorded perfectly, and temporal information on
the spread of the disease is rich. This allows the fitting of fairly
sophisticated stochastic models for how the disease spreads from host to
host, incorporating such effects as the increasing resistance plants
develop as they age, using Markov chain Monte Carlo techniques and
augmenting the parameter space. We discuss issues of model selection,
and present a way of using stochastic residuals as a means of
intuitively choosing from competing models. We believe that the
techniques presented can be generalised to other problems to extract
biologically useful information.
In this talk, we will analyze a queueing system characterized
by a space-time arrival process of customers served by a countable set of
servers. Customers arrive at some points in space and the server stations
have space-dependent processing rates. The workload is seen as a Radon
measure and the server stations can adapt their processing allocation to
the current workload. We derive the stability region of the queuing system
in the usual stationary ergodic framework.
From the analysis of this stability region, we define optimal partitions
of space among server stations. Wireless communication networks provides
a natural field of application for this model.
Some specific subclasses of policies are also studied and we will we give
extensions to more complex systems.
Let $\zeta_1,\zeta_2,\ldots$ be independent random variables,
$$
Z_n=\sum_{i=1}^n\zeta_i,\qquad \overline{Z}_n=\max_{k\leq
n}Z_k,\qquad Z=\overline{Z}_\infty.
$$
It is well known that if $\zeta_i$ are i.i.d. random variables
then $Z<\infty$ a.s. when $a=-\mathbf{E}\zeta_i>0$, and $Z=\infty$
a.s. when $a=0$.
In applications dealing with stochastic models of random walks
$\{Z_n\}$ with small $a>0$ (heavy traffic problem in queueing
theory, ruin problem for insurance companies with low income and
others) an important question is how large is $Z$ and what is the
limiting distribution of normalized $Z$ as $a\to 0$. The answer to
this question in case when $\zeta_j$ are i.i.d. random variables
and $\sigma^2=\mathbf{E}\zeta_j^2<\infty$ is well known:
$\mathbf{P}(aZ>t)\Longrightarrow e^{-\frac{2t}{\sigma^2}}$ as
$a\to 0$.
The talk is devoted to the study of the limit distribution of
$\overline{Z}_n$ in an important in applications case when
$\mathbf{E}\zeta^2_i=\infty$ and $\zeta_i$ are non identically
distributed, $a=1/n\sum\limits_{i=1}^na_i\to~0$. The function
$d(a)$ has been found such that for $n=\frac{Td(a)}{a}$
($T\leq\infty$ is independent of~$a$) there exists the limit
distribution of $\frac{Z_n}{d(a)}$. If $T=\infty$ then for some
cases this limit distribution has been found in an explicit form.
We analyse a financial contract that is traded in the UK gas industry
(and also the US and other deregulated gas industries) that entitles the
contract holder to multiple (sequential) exercise rights in a finite time
interval. The derivative product, called a swing option,
is treated as an American--type option but with multiple stopping
times. It is valued by adapting a version of the least squares Monte
Carlo algorithm (Longstaff \& Schwartz (2001)). Since
this algorithm involves regression, we will also discuss the
appropriate choice of basis functions and other computational issues.
The talk will conclude with sample results for the swing option
calibrated for the UK gas market.
I am going to give an overview on several models of random graphs and
random graphs processes together with basic methods for studying such
structures. Applications to cluster analysis, cryptology, modelling
epidemic processes, analysis of algorithms will be discussed as well.
We study arbitrage opportunities in diverse markets as introduced by R.
Fernholz. In this context, a market is said to be 'diverse' if no stock
is ever allowed to dominate the entire market in terms of market
capitalization. By a change of measure technique we are able to generate
a variety of diverse markets. The construction is based on an absolutely
continuous, but non-equivalent measure change which implies via the
optional decomposition theorem the existence of instantaneous arbitrage
opportunities. For this technique to work, we single out a crucial
non-degeneracy condition. Moreover, we discuss the dynamics of the price
process under the new measure as well as further applications.
It is well-known that in the standard Black-Merton-Scholes market model
(BMS) stock prices are modelled as geometric Brownian motion. On the other
hand, market data show that returns differ from this benchmark. Asset
prices jump, leading to non-normal and heavy-tailed distributions, and
return volatilities vary stochastically over time, reflecting incomplete
financial markets. In relation to an investigation of better fitting stock
price models, which already started with Mandelbrot (1963), the class of
L\'{e}vy processes including Brownian motion and the (compound) Poisson
process reveal a more realistic image of the stochastic structure of asset
prices.
Recalling important semimartingale properties of (exponential) L\'{e}vy
processes, we consider the - still open - problem how we can transfer a
market price of risk approach of the complete BMS model to incomplete
market models (like e.g. the market price of credit risk of Giesecke and
Goldberg). A completion of this approach would allow us to construct
explicitly equivalent local martingale measures with the help of suitable
density processes. A detailed analysis of the jumps of such density
processes reveals the importance of the stochastic logarithm and its
impact on change of measures techniques in the sense of Girsanov including
weak and strong predictable represenation properties. We then ask for
applications of these techniques to L\'{e}vy processes.
Any non-negative monotone and integrable function admits
an integrable majorant which is regularly varying at infinity.
We give a sketch of the proof of this result (the result seems to
be new!) and provide a number of applications. In particular, we
formulate a new criterion for transience of Markov chains which
is ``unimprovable'' in a certain sense.
Let us suppose that our observed data consist of the sample path of
the process Y on a continuous time horizon [0,T]. We assume that
the observation process Y has the dynamics
dY_t=X_tdt +dw_t,
where w is a standard Brownian motion and the signal process X
depends on an unknown parameter \theta. Our objective is the
estimation of this parameter. Here are some examples:
1. X_t=S(\theta, Y_t), so X is adapted to the filtration
generated by Y;
2. X is an Ornstein-Uhlenbeck process, independent of the Brownian
motion w;
3. X is a Markov process having a finite state space, independent
of w (called a Hidden Markov Model).
We are going to address the questions of consistency and asymptotic
normality of the Maximum Likelihood Estimator (MLE) as the observation
time T goes to infinity.
Important features of survival data concerning onset of inherited
disorders are (i) penetrance of the mutation causing the disorder,
meaning that presence of the mutation need not confer 100% risk
of suffering the disorder; (ii) ascertainment bias, arising because
the families studied are often those with unusually severe histories
of the disorder; and censoring which is present as usual.
Epidemiologist have specified different models of the process of inherited,
censoring mechanism and ascertainment scheme to estimate parameters.
Retrospective studies commonly may not controlled one or more of the
latter arising a misspecification problem. Gui & Macdonald (2002)
suggested a Nelson-Aalen estimate for a certain
function of the rate of onset. We will review somegenetic epidemiology
work and analyse some properties of an extension of Gui & Macdonald (2002)
underlying model that allow us to obtained some estimators.
In this talk, the elegant method of Dynamic Programming (DP) will be
introduced in a non-technical way, and extensions will be considered for
when DP is not immediately applicable. DP is a
computationally-efficient method for finding the global solution to some
optimisation problems. For example, it can be used to track boundaries
in order to automatically segment 2-D medical images into different
anatomical regions (Glasbey and Young, 2002). It can also be used to
align pairs of tracks in 1-D electrophoresis gels, using the method of
Dynamic Time Warping which is also used in automatic speech recognition.
However, if images are three dimensional, or many gel tracks need
aligning, then simple DP is not possible. Extensions to DP will be
considered, illustrated by applications in 3-D X-ray computed tomography
and pulsed field gel electrophoresis.
Glasbey, C.A. and Young, M.J. (2002). Maximum a posteriori estimation
of image boundaries by dynamic programming. Applied Statistics, 51,
209-221.
A new valuation system(NUMAT) has been suggested---life insurance
policies are valued based on a replicating portfolio. Under NUMAT,
Margrabe options can be used to ensure solvency while at the same time
giving the insurer greater investment freedom. Traditional policies
and convetional with-profits policies have been studied and different
strategies are proposed to see how solvency can be achieved in this
new valuation system.
TBA
In this talk we will discuss various issues relating to
the securitisation of mortality risks. This will cover some
basic issues related to contract design including comment
on the bonds issued by Swiss Re and BNP Paribas.
We will also discuss the different approaches that might be
taken to modelling stochastic mortality by drawing from the
established field of interest-rate modelling. Amongst these
we will focus on the Annuity Mortality Market Model
and the SCOR Market Model.
The application of extreme value theory (EVT) methods to time series of
financial returns has been a subject of interest in recent years. Most
studies have focussed on applying static tail estimation techniques
under assumptions of stationarity, such as the Hill estimator or the
generalized Pareto tail approximation method. The aim of this talk is to
propose a new dynamic model for the occurrence of extremes above some
high threshold in a financial time series. The model attempts to
describe both the temporal occurrence and the magnitude of threshold
exceedances and does so by employing a point process formulation with
self-exciting structure and a parameterization inspired by standard EVT
models. The model is applied to financial data and used to estimate a
stylized Value-at-Risk (i.e. an extreme quantile of a conditional return
distribution for the next time period).
We study the asymptotic behaviour of the moments of ladder height and
ladder epoch for a random walk with heavy tailed distribution of summands.
We assume that the drift of the walk tends to zero.
The talk contains some upper bounds for the moments of ladder values
and a limit theorem for the expectation of the ladder epoch.
In recent years, there has been a lot of activity in modeling the
Internet at various levels. Two examples are: (a) Modeling at the packet
level has resulted in a number of interesting phenomena
(long-range dependence and heavy tails) the consequences of
which are a subject of intense research;
(b) Modeling at the flow level, i.e., at a more macroscopic
level has also resulted in important performance and operational
observations, such as stability. In this talk, we will describe
on the latter model. In a sense, the model resembles the classical
circuit switched (loss) network model of Telephony, but differs from it
in that a connection is never blocked. Rather, it is
admitted at the expense of degradation of service. The resulting
stochastic process can be thought of as a multi-dimensional
Markov chain in continuous time. The stability is studied by
means of a Lyapunov function. Interesting questions arise
when we depart from the Markovian assumption, or study related
models. For instance, it is still unknown whether stability of
the fluid model implies stability of the original system
under non-Markovian assumptions. We will focus on existing
and future research on this topic.
In this talk I address the problem of sampling
colourings of a graph~$G$ by Markov chain simulation.
Mostly, the ``colorings'' will be the usual proper
colourings of~$G$, but I'll also touch on more general
``$H$-colourings''. (In statistical physics terms,
I'll be considering spin systems with hard constraints,
with particular emphasis on the antiferromagnetic Potts model.)
There is a substantial
body of literature concerned with bounding {\it mixing time\/}
(i.e., time to convergence to near-stationarity)
of Markov chains defined on colourings of a graph~$G$.
Almost all
this theoretical work relates to random single-site
updates, which choose a random vertex for updating
at each transition. We shall refer to this strategy
as {\it Glauber dynamics}. However, experimental work
is often carried out using systematic strategies that
cycle through coordinates in a deterministic manner,
a dynamics we refer to as {\it systematic scan}.
The mixing time of systematic scan seems more difficult
to analyse that that of Glauber, and little is currently
known. I'll describe some early steps in the rigorous
analysis of systematic scan.
The work I describe was done jointly with Martin Dyer
and Leslie Goldberg.
In his seminal paper from 1978, Sheldon Ross set up few
conjectures which formalize a common belief that more variable arrival
processes lead to worse performance in queueing systems.
In the talk we will introduce two type of orderings
$\le_{\rm idcx}$ and $\le_{\rm idcv}$ of random vectors,
which in turn yields respective orderings of stochastic processes
and show their relevance for studying results
of the above type.
We also mention an application to risk theory.
Insurance risks can be embedded in bonds to create
insurance-linked securities. Catastrophic property risks have been
transfered to bondholders in this way very successfully, beginning in
the early 1990s. I will show how mortality risks can be securitized in
a similar way.
In this talk we consider methods of calculating and
approximating the density and the moments of the time to ruin in the
classical risk model. In particular, we will indicate how the density of
the time to ruin can be obtained through Laplace transforms.
It is well known that stock returns on short time horizons
are highly non-normal, contrary to the assumptions in the Black-Scholes
model. This has important implications for option pricing. Cerny(2002)
shows that the risk premium associated with the size of optimal hedging
errors in a realistically calibrated multinomial lattice can account
for much of the discrepancy between the historical and implied volatility.
The lattice calculations of hedging error tend to be computationally
intensive, particularly for long times to maturity and very short
rehedging intervals. The present paper overcomes that difficulty
by computing the hedging error in the continuous-time limit of the
multinomial lattice. This is done by means of a Fourier transform of
the mean value process, which permits fast computation regardless of
the time to maturity. The paper provides an efficient implementation
of the hedging error formula via FFT and examines its speed and accuracy.
Joint survival analysis plays an important role in such actuarial
applications as pricing joint life policies. Last survivor insurance has
become a major phenomenon in the U.S. as a result of the aging population.
Last survivor policies are generally used by older couples for estate tax
purposes and carry large amounts of insurance.
As demonstrated by Parkes et al. (1969), Pruitt (1993), Luff and Vose
(1994), there is an evidence of strong statistical dependence between the
future lifetimes of the insured spouses. Ignoring this dependence may bring
about a substantial under-pricing of the joint last survivor policies.
Frees et al. (1996) introduced copula models to the construction of joint
mortality functions. Shemyakin and Youn (2001) used Bayesian approach
allowing for incorporation of prior information on individual mortalities.
Youn, Shemyakin and Herman (2002) re-examined this construction. Further
progress in model-building is presented. The models developed are applied to
a database of approximately 15,000 joint annuity contracts from a large
Canadian insurer.
Consider a random walk $S_n=\xi_1+...+\xi_n$ with i.i.d. increments
and assume that these increments are heavy-tailed. The exact
asymptotics for the tail distribution of the supremum $M=\sup_{n\ge
0}S_n$ and for tha tail distribution of the maximum
$M_\tau=\max_\{0\le i \le \tau\} S_i$ on the random time interval
$\tau=\min\{n\ge 1: S_n\le 0\}$ are known when $E\xi_1$ is negative
and finite. We revisit these theorems and give complementary results
in the case $E|\xi_1|=\infty$.
The talk is dedicated to localization of the principal eigenvalue (PE)of
the Stokes operator under the Dirichlet condition on the boundary of a
fine-grained random domain contained in a large cubic block. The
random microstructure is assumed essentially independent and
identically distributed in distinct unit cubic cells. As the volume of
the containing block goes to infinity, the PE exhibits deterministic
behaviour. It converges to zero at the same rate as the corresponding
quantity for a domain that is obtained by dilation from a set of fixed
shape
and has volume proportional to the logarithm of that of the containing
block.
The main result in this talk is a theorem extending from the planar
case
to higher dimensions the author's earlier result on convergence of the
appropriately normalized Stokes PE to a non-random limit in
probability.
It develops a new approach to the study of deterministic asymptotics
of the
PE that is based on recent work of F.Merkl and M.V.Wütrich [3].
Localization of the principal eigenvalue (PE) of an elliptic operator
with
random elements acting on functions in a standard domain of very large
volume
attracts considerable interest since mid-eighties. Rigorous research
in this field,
which remains active, was started by A.-S. Sznitman (see, e.g., [1]).
The investigation originated in physics of disordered media: the PE of
the
Laplacian in a domain with random fine-grained boundary carrying zero
Dirichlet
condition determines the rate at which diffusing particles are
absorbed by
randomly positioned traps.
Known versions of the "greyscale" techniques originating in work of
A.-S. Sznitman exploit the possibility to exclude from the domain
those parts where the boundary is massively present (see, e.g., [1,2]
and the references therein; this approach remains efficient when the
boundary is substituted by a random positive potential). The
restriction to the effective domain is done on the basis of a
selection rule, which identifies the "vacuities" consisting of cubic
cells where the boundary (or potential term) is inessential. The
maximal volume of a connected "vacuity," which determines the PE, is
estimated by techniques used in the percolation theory to analyse
random "lattice animals." An additional difficulty in the case of the
Stokes operator is the necessity to use only divergence-free test
functions.
The localization of the PE in [3] for the Schrödinger operator with a
small
positive random potential is based on the analysis of feasibility of
specific
values of the Rayleigh quotient for individual test functions. The
results of
[3] include a description of transition from the limiting PE values
for the small
random potential to those appearing in the original problem with
random boundary
(or a "large" potential term).
The new approach suggested in [3] proved efficient in the derivation
of the
lower bound on PE also for the Stokes operator [4], which is discussed
in the
present talk. In [4], a lower bound for the Stokes PE is derived
through low
compressibility approximation using methods of [3], and the
corresponding upper
bound is obtained by construction of a test function with low
Rayleigh quotient
that is compatible with a typical configuration of random structure
using an
argument inspired by the cited "greyscale" techniques.
REFERENCES
[1] Sznitman A.-S. Brownian Motion, Obstacles and Random
Media. Springer-Verlag,
New York, 1998.
[2] Yurinsky V.V. Localization of spectrum bottom for the Stokes
operator
in a random porous medium. Siber. Math. J. (2001) vol.42, No.2,
386-413.
[3] Merkl F., Wütrich M. Infinite volume asymptotics of the ground
state energy
in a scaled Poissonian potential. Ann. Inst. H.Poincaré. Probability
and
Statistics (2002) v.38, No 3, 253-284.
[4] Yurinsky V.V. Localization of the Principal Eigenvalue for the
Stokes
Operator in a Random Domain. Depto de Matemática - Centro de
Matemática,
Universidade da Beira Interior, Pré-publicação No 1, 2003. Available
at
http://www.ubi.pt/externos/noe.html
Equilibrium asset pricing models provide a framework for studying ways
in which risk is exchanged among agents in a competitive market and for
calculating the prices of traded risks. In the actuarial literature,
Buehlmann's (1980, 1984) classic equilibrium models provide transparent
formulas for risk allocations and prices. In this investigation, we
consider equilibrium asset pricing models where agents operate under a
distorted probability. Distorted probabilities are used to represent rank-
dependent preferences (Quiggin, 1993), Knightian uncertainty (Schmeidler,
1989) and risk measures (Wang, 1996). The solution of such equilibrium
models requires a way of aggregating preferences in the case of distorted
probability, which is achieved in this study by the definition of a
"collective ambiguity aversion" coefficient. Explicit pricing formulas
are obtained, which can be seen to be generalisations of the ones derived
by Buehlmann (1980, 1984). Our formulas feature an additional term due
to probability distortion, which tends to inflate asset returns (or
equivalently increase the price of insurance). The effect of probability
distortion on agents' trading strategies is manifested by their developing
a "betting behaviour" against other agents' beliefs.
As a special case of the previous models, we obtain an equilibrium
model where agents' objective functions are given by distortion risk
measures. It is shown that a necessary condition for equilibrium is that
agents identical risk measures. Given the comonotonicity of equilibrium
allocations, this means that if a regulator imposes the same risk measure
on all market participants, they will tend to make similar investment
decisions, which may in turn increase the likelihood of a systemic crisis.
Finally, we show that, in the framework of this model, asset prices are
consistent with those implied by risk capital allocation models based
on cooperative game theory.
We consider stochastic epidemic models in which population is divided
into local groups, and in which mixing can occur both locally within
groups, and also between groups. Statistical inference for such models
is complicated by the fact that the likelihood of the final outcome
(i.e. numbers infected) is intractable. Two methods to overcome this
are described:
(i) an approximation using the limiting behaviour of the model in a
large population;
(ii) a non-approximate method using a random graph representation to
describe the actual spread of infection.
Both approaches are implemented using Markov chain Monte Carlo methods,
and for illustration are applied to data on influenza outbreaks.
Low frequency, high severity, events present an important challenge to
insurance risk management. An insurer's ratings, and indeed solvency,
depend on the rigour with which catastrophe risk is handled. The past
decade has witnessed major developments in the provision of computer
software for loss aggregation and probabilistic risk analysis of insurance
portfolios. The domain of analysis extends from earthquake and
windstorm to terrorism and mortality catastrophes. A review is given of
these developments, which are of both intellectual and commercial interest
to actuaries.
We give explicit upper bounds for convergence rates when approximating
(both one- and two-sided general curvilinear) boundary crossing
probabilities for the Wiener process by similar probabilities for close
boundaries (of simpler form for which computing the probability is
feasible). In particular, we generalize and improve results obtained by
P'otzelberger and Wang (2001) for the case when approximating boundaries
are piecewise linear. Applications to barrier option pricing are discussed
as well.
First a systematic methodology is developed for the solution
of the pricing of bonds (including coupon bonds, and options on
such bonds), based on the concept of small perturbation theory, in which
the volatility of interest rates is treated as a small parameter,
thereby permitting a series expansion procedure to be developed.
The method is applicable to a broad class of stochastic interest-rate
models, including those of Cox, Ingersoll and Ross (CIR) and Vasicek.
Prototype calculations indicate a high degree of accuracy of the simple
formulae thus obtained (appropriate for use with a hand calculator)
when compared with calculations using the more detailed full equations.
Second, the methodology is applied to an option-pricing problem of practical
interest. With the advent of genetic testing to screen for disease
susceptibilities, there exists the possibility for individuals
to use this information to take out insurance products that maximize their
potential return from insurance policies that are calculated from life tables.
This talk proposes a method for amalgamating medical insurance and
superannuation (pension) funds using financial derivatives (priced using
the simple formulae derived in the first part of the talk)
to counter adverse selection. Using a combination of derivative products, the
risk of adverse selection is hedged during the life of the client,
while preserving the level of insurance, without necessarily
increasing the price.
This presentation summaries the results of author's six-year research
work at the Center on Aging, University of Chicago, aimed to explore
determinants of human lifespan (several projects supported by the National
Institute on Aging, USA).
(1) The effects of parental age at person's conception on person's
lifespan are studied in a context of mutation theory
(2) The effects of person's month-of-birth on person's lifespan
are studied in a context of 'fetal origins of adult disease' concept
and the idea of early-life seasonal programming of adult lifespan.
(3) The effects of parental lifespan on person's lifespan
are studied in a context of genetics of aging and genetics of quantitative
traits.
(4) We also tested widely publicized claims published in Nature (1998),
that human longevity comes with high cost of infertility (half of
long-lived women were reported to be childless).
(5) Finally, if time permits, we will discuss possible explanations of
aging and longevity in terms of reliability theory.
I will discuss time-homogeneous random walks on two-dimensional
complexes. A two-dimensional complex is a union of a finite number of
quarter plane lattices connected at one dimensional boundaries. I will
consider the specific case where each boundary belongs to only two
quarter planes. All of the results are formulated in a constructive
way. By this I mean that for any given random walk we can, with a
concrete calculation using the first and second moments of the jumps,
conclude whether the process is recurrent or transient. The main new
result is for a critical case where the long-term behaviour of the
random walk is very similar to that found for walks with zero mean
drift inside the quadrants even though this walk has non-zero drifts.
The ideas can be applied directly to two-dimensional exhaustive
polling models in critical cases. I will also discuss the non-critical
many dimensional model, see S.Foss, G.Last,
(Ann.Appl.Probab.,1996).
A paper, joint with Mikhail Menshikov, with these
results is soon to appear in Ann.Appl.Probab.
The following two questions are examples of standard
questions which many pricing actuaries are confronted with.
- own data versus industry wide data.
Often, own company data as well as industry-wide data are available. But
how much should one rely on the own data and how much on industry-wide
date when calculating a tariff ?
- "normal" claims and "big" claims In many lines of business, a
small number of bigger claims (only 1% or 2% of the total number of
claims) make more than half of the total claims load. How should we
calculate the pure risk premium corresponding to the big claim part
based on rather few observations ?
The appropriate actuarial technique and answer to many of such questions
is multi-dimensional credibility. There are two reasons for this:
- credibility is particularly suited in cases with little data
- multidimensional credibility takes simultaneously the observations
from the different categories into account and lets the data tell us,
whether and how much we can learn from the one category with respect to
the other.
In the seminar, the multidimensional credibility model is presented and
the corresponding credibility estimator is derived. Next the methodology
is applied to a real data set from motor insurance to estimate the
frequency of big claims. At the end, a general result concerning
optimal data compression is presented and it is shown, that
multidimensional credibility also covers the credibility regression
case.
On 1 January 2002 new life insurance valuation rules
were introduced in Denmark. Here the emphasis is on determining a
market based liability of the guaranteed benefits of participating life
insurance contracts. The paid-up benefit valuation method plays an
important role in the new Danish market based life insurance valuation
rules. In Linnemann (2000, 2002, 2003a) we present the theoretical
background to the paid-up benefit valuation method for level premium
paying participating life insurance contracts. Moreover in Linnemann
(2002) we give a theoretical basis for amendments and further
developments of the new Danish market based life insurance valuation
rules. We suggest that the so-called extended paid-up benefit valuation
method should be used for the valuation of the guaranteed benefits of
participating life insurance contracts. The cash-flows that should
enter in the calculation of a market based value of the guaranteed
benefits of participating life insurance contracts are being determined
by this method. We point out that the valuation principles of `coherence
between the benefits and premiums being valued' and `avoidance of future
valuation strains' are relevant in a market based valuation regime. In
Linnemann (2003b) we review the above papers.
We present in the lecture a number of the principal considerations and
results that have been dealt with in the above papers.
Linnemann, P. (2000). An actuarial analysis of participating life
insurance. Pen-Sam, Working Paper, August 2000. Published in
Scandinavian Actuarial Journal 2003, 153-176.
Linnemann, P. (2002).
Valuation of participating life insurance liabilities. Pen-Sam, Working
Paper, April 2002. To be published in Scandinavian Actuarial Journal.
Linnemann, P. (2003a). An actuarial analysis of participating life
insurance. Scandinavian Actuarial Journal 2003, 153-176. Errata 177.
Linnemann, P. (2003b). Market based valuation of guaranteed benefits of
participating life insurance contracts. Pen-Sam, Working Paper, June
2003.
Actuarial valuation should be understood in a
multidimensional sense. In practice this means that the actuary should
express the liabilities of the insurer as a portfolio of financial
instruments. This Valuation Portfolio can be calculated policy-wise. For
risk management purposes the aggregated valuation portfolio has to be
compared with the investment portfolio.
In this talk we consider a continuous time Markov process
killed at the exit time T from a subset A of its state space.
We assume that T is finite a.s. We define the concept of
Never Exiting (NE) Markov process from A.
The first question is to determine
when this process exists and the second is whether we can define
it by the change of probability measure argument.
It turns out that NE process is
Markovian and we will study its properties. In particular we give
the relationship between the stationary distribution of NE Markov
process and the quasi-stationary distribution. The
general scheme can be found in the papers Jacka and Roberts (1995)
and Lambert (2000). We apply the results to a workload process of
G/G/1 queue conditioned to stay positive. We consider the cases
when the service times distribution is light-tailed and regularly
varying.
An oscillating random walk (whose increments have second moments) when
conditioned to stay positive may be seen in some sense as an analogue to a
Bessel-3 process; since a Bessel-3 is also equal in law to a Brownian motion
conditioned to stay positive. Like Brownian motions, Bessel-3 processes obey
LILs at large times. It is therefore natural to ask if, like random walks with
second moments, an oscillating random walk conditioned to stay positive also
obeys an LIL at large times.
Using three fundamental facts: 1) the Bertoin-Doney description of the step
distribution of conditioned random walks, 2) Tanaka's fundamental path
decomposition of conditioned random walks and 3) a new Skorohod-type embedding
of conditioned random walks in Bessel-3 processes, we establish
an LIL result as
well as LIL-type results for the slowest growth rates.
This is joint work with G. Kersting (Frankfurt) and Ben Hambly (Oxford).
Consider a random walk $S_n=\xi_1+...+\xi_n$ with i.i.d. increments.
S. Asmussen has found the exact asymptotics
for tha tail distribution of the maximum
$M_\tau=\max_\{0\le i \le \tau\} S_i$ on a random time interval
where $\tau=\min\{n\ge 1: S_n\le 0\}$. He assumed the condition
$-\infty< E\xi_1<0$ to hold.
We give a complementary result in the case $E|\xi_1|=\infty$.
Let $\{\xi_i\}$ be a sequence of i.i.d. random variables and
$S_n = \sum_{i=1}^n \xi_i$.
For two classes of subexponential distributions,
we obtain new uniform upper bounds for the
ratios $P(S_n > x) / P(\xi_1 > x)$.
Then we apply these bounds to the asymptotic study of a Markov-modulated
random walk with heavy-tailed increments.
Traditionally the cost of the guarantees under UK unitised with-profit
policies has been ignored or priced in a very imprecise way. However
in this paper I will charge the policyholder for these guarantees an
amount equal to the price of matching put options. Using simulations I
show how the distribution of payouts compares under different levels
of guarantees. Further I investigate the size of the free estate if
charges are deducted according to option prices but the estate is
actually invested in bonds.
The focus of this talk is on peer-to-peer systems, and more precisely on
what peer relations to maintain in such systems. Those peer relations
are naturally modelled as a graph, and one question of interest is how
to create or adapt such a graph so as to meet desired reliability
objectives, e.g. that connectivity be retained with a given proportion
of random link failures. One constraint specific to the context of
peer-to-peer systems is that graph adaptation should involve distributed
/ local operations only.
I will describe local rules for adapting a given graph so as to improve
its reliability, and an analysis of the resulting graph's connectivity
properties. If time allows I will also discuss a somewhat related issue,
namely how to sample uniformly from the node set of a graph.
It is well known that in an incomplete financial market, the super
replication price of a contingent claim coincides with the supremum of
its expected values over the set of pricing measures. For the case of a
contingent claim which involves possibly unbounded losses however, super
replication using only admissible trading strategies would lead to a gap
between the interval of prices and the super replication price - the
choice of permissible strategies becomes crucial.
Consider a financial market in which an agent is permitted to trade with
only utility-induced restrictions on negative wealth. For a sufficiently
integrable (but possibly unbounded) contingent claim, we give a
representation of the utility-based super-replication price of the claim
as the supremum of its discounted expectations under pricing measures
with finite generalised entropy.
Central to the proof of this result is a bipolar relation between the
cone of super replicable contingent claims with zero initial endowment,
and the cone generated by pricing measures with finite loss-entropy.
A method is presented for defining a dynamic risk measure from a static
risk measure using backwards iteration. This method is applied to the
CTE risk measure to produce the iterated-CTE (ICTE). It is shown that
the ICTE is coherent, consistent and relevant. Formulae for the ICTE
when the loss process is lognormal are shown. Implementation of the ICTE
to equity-linked insurance with maturity and death benefit guarantees is
discussed.
When modelling change in animal abundance, empirical modellers typically
ignore the population processes. Hence they can obtain estimated rates of
change that are biologically implausible, and they have no mechanism for
predicting the effects of different management strategies, or for
modelling movement between components of a metapopulation. Conversely,
mathematical modellers have typically failed to integrate fully
stochasticity in the population processes, and uncertainty in estimates of
population parameters. In this talk, we show how state-space models
provide a framework for embedding stochastic population dynamics models
fully into inference. A framework is also presented that allows complex
models to be constructed as a sequence of simple process models.
We consider a situation originally discussed by De Finetti in
1957 in which an insurance surplus process is modified by the introduction
of a constant dividend barrier. We extend some known results relating to
the distribution of the present value of dividend payments until ruin in
the classical risk model and show how a discrete time risk model can be
used to provide approximations when analytic results are unavailable. We
extend the analysis by allowing the process to continue after ruin. We
also provide an extension of De Finetti's results by considering the
binomial-geometric risk model.
I shall discuss recent progress in the theory of randomly
forced 2D Navier-Stokes equations and the relevance of
these results for statistical hydrodynamics. I assume
"almost no" knowledge of PDEs and almost no knowledge of random
processes.
We consider a multi-server queue with FCFS service discipline.
Under various conditions, convergence rates of the workload
process to the stationary one have been obtained.
Proofs are based on the renovating events method and on the
saturation rule techniques. By comparison with a single-server
queue, we show that our results are unimprovable (in a certain sense).
GAMs constructed using basis functions offer advantages over backfit GAMs
in terms of model selection and inference. In particular such GAMs are
simply penalized GLMs, which tends to lead to quite straightforward
fitting and inference methods so long as some method can be found for
estimating the appropriate degree of smoothness for each model term.
However, efficient selection of the degree of smoothing is heavily
dependent on using relatively low rank bases for smoothing. In this talk I
will discuss the production of low rank smoothers designed to meet certain
optimality criteria: namely that given their rank the smooths are in some
sense as close as possible to an equivalent full thin plate spline
smoother. I will also discuss methods for selecting the degree of
smoothing in GAM models, and may also touch on how to obtain GAM
confidence intervals with good coverage properties. The methods discussed
are implemented in R package mgcv. One or two applications to fisheries
data will be presented.
I shall discuss statistical properties of a family of dynamical
systems acting in the space of integer valued sequences, which
model dynamics of simple deterministic traffic flows. Applying
ideas borrowed from substitution dynamics we are able to reduce
the analysis of the traffic flow models corresponding to the
multi-lane traffic and to the flow with fast particles (with
velocities greater than 1) to the simplest case of the flow
with the one-lane traffic and slow particles, where the crucial
technical step is the derivation of the exact life-time for
a given cluster of particles. Applications to the optimal
redirection of the multi-lane traffic flow and a model of a
pedestrian going in a slowly moving crowd will be discussed
as well.
Conditional expected values in Markov chains are
solutions to a set of associated backward differential equations, which
may be ordinary or partial depending on the number of relevant state
variables. We investigate the validity of these
differential equations by locating the points of non-smoothness of the
state-wise conditional expected values, and present a numerical method
for computation of such expected values with controlled
global error. Three cases leading to first order partial differential
equations in two variables
are considered, all from finance and insurance: Option pricing
in a Markov chain driven financial market; Probability distributions
of cash flows generated by multi-state life insurance contracts;
Reserves in life insurance when payments or intensities are path-dependent.
(pdf-file at http://stats.lse.ac.uk/norberg Recent Papers)
We study a probabilistic model of a computer processors system performing
large-scale parallel simulations. The model is defined in terms of
interacting particle systems. Every particle in the system has it's own,
''free'' dynamics, which is a one-dimensional simple random walk. Besides,
there is a mean-field type interaction between particles synchronizing the
positions of the particles on the line.
We prove that there exists so called hydrodynamical limit of the
process describing distribution of the particles on the line as the number
of particles infinitely grows. The hydrodynamical limit is a
deterministic process and is defined as a solution of some partial
differential equation. We distinguish the cases when every particle
has non-zero or zero drift generated by ''free'' dynamics.
As usually, the hydrodynamical scaling and limit depend on the value
of the free dynamics drift. It is interesting to note that the partial
differential equation arising in the case of zero drift is a famous
Kolmogorov-Petrovski-Piskunov (KPP) equation.
We discuss also connections between asymptotic behavior
of the interacting particle system and properties of
solutions of the limiting PDE-equations.
(joint work with Anatoli Manita, Moscow State University)
We see measurement of multiperiod risk as assigning to "value"
processes
"risk-adjusted value" processes, in a coherent way.
Like in the one-period case, test probabilities provide the main
tools,
but theycan be used in two ways:
- directly, at initial date and later with conditional expectations
- with backward induction.
Generalisation of Snell envelope construction and the Bellman's
principle help to characterize the so-called "stability",
"time consistency" (T. Wang), "rectangularity" (L. Epstein and
M. Schneider)
properties of the set of test probabilities and their equivalence.
This property ensures that the two construction are the same.
(joint work with F. Delbaen, J.-M. Eber, D. Heath and H. Ku)
In this talk, we focus on the cost of data dissemination from one source
to many destinations in a communication network represented by a random
oriented tree. The multicast mode is characterized by the ability of
some vertices to replicate a received packet depending on the number of
destinations downstream. The two groups of stochastic assumptions --
that the trees are generated by a Galton-Watson process or by point
aggregates of a spatial Poisson process -- are meant to represent tree
shapes arising in the wired and wireless parts of the Internet. We are
interested in the impact of multicast on the overall traffic volume and
other tree-related cost functions, which we evaluate using traffic
conservation laws and classical technique of branching processes.
This is a joint work with B.Blaszczyszyn (INRIA-ENS).
Using the Wiener-Hopf factorisation it is shown that it is possible to
bound the path of an arbitrary Levy process above and below by the
paths of two random walks. These walks have the same step
distribution, but different random starting points. In principle, this
allows one to deduce Levy process versions of many known results
about the large-time behaviour of random walks. This is illustrated by
some results about Levy processes which converge to infinity in
probability.
In this paper we derive a market value for Guaranteed Annuity Option using
martingale modelling techniques. Furthermore, we show how to construct a
static replicating portfolio of vanilla interest rate swaptions that
replicates the Guaranteed Annuity Option. Finally, we illustrate with
historical UK interest rate data from the period 1980 until 2000 that the
static replicating portfolio is extremely effective as a hedge against the
interest rate risk involved in the GAO, that the static replicating
portfolio is considerably cheaper than up-front reserving and also that
the replicating portfolio provides a much better level of protection than
an up-front reserve.
In this talk we discuss a new family of term-structure models
based upon the Flesaker & Hughston (1996) positive-interest
framework. We demonstrate that, besides being suitable for
derivative pricing, the models are ideally suited
for use in long-term risk management. In particular, the models
can be parametrised in a way which gives sustained periods of both high
and low interest rates, similar to the cycle lengths we have
observed over the course of the 20th century in the UK and US.
The accompanying paper is online at
http://www.ma.hw.ac.uk/~andrewc/papers/ajgc30.pdf
Conditional expected values in Markov chains are
solutions to a set of associated backward differential equations, which
may be ordinary or partial depending on the number of relevant state
variables. We investigate the validity of these
differential equations by locating the points of non-smoothness of the
state-wise conditional expected values, and present a numerical method
for computation of such expected values with controlled
global error. Three cases leading to first order partial differential
equations in two variables
are considered, all from finance and insurance: Option pricing
in a Markov chain driven financial market; Probability distributions
of cash flows generated by multi-state life insurance contracts;
Reserves in life insurance when payments or intensities are path-dependent.
(pdf-file at http://stats.lse.ac.uk/norberg Recent Papers)
The sensitivity of a price function to changes in the model
parameters is given by its derivatives w.r.t. the parameters -
the so-called Greeks. The Greeks are easily calculated when the
price possesses a closed form expression. In any case we may compute
the Greeks as solutions to differential equations derived from
the differential equation of the price function by simply
differentiating it w.r.t. the parameters. To prove the existence of
the dynamic Greeks is the hard part. The idea extends to other dynamic
entities. Some examples with numerical illustrations are given, both from
insurance and finance.
It is generally acknowledged that the molecular mechanisms regulating
key cellular processes such as gene expression are intrinsically
stochastic. The random diffusion of cell signalling molecules and the
combinatorial assembly of transcription factor complexes provide
extensive opportunities for the action of chance. In recent years
stochastic regulatory network models have been developed, based on
discrete-event simulation techniques for generating realisations from
the complex continuous-time countable-state Markov processes governing
the reaction systems. These models contain many parameters with
uncertain values. In addition, the latent process can only be observed
partially, and at discrete time intervals. Inference for such Markov
process models is an extremely challenging problem.
This paper investigates option prices in an incomplete stochastic
volatility model with correlation. In a general setting, we prove an
ordering result that convex option prices are decreasing in the market
price of volatility risk.
We investigate the q-optimal class of pricing measures. Using the
ordering result, we prove comparison theorems between option prices under
the minimal martingale, minimal entropy and variance optimal pricing
measures. If the mean-variance tradeoff is deterministic, this collapses
to the well known result that option prices computed under these three
pricing measures are the same.
Specialising to the Heston model with mean-variance tradeoff increasing
in volatility, enables us to deduce option prices are decreasing in the
parameter q. Numerical solution of the pricing pde corroborates the theory
and shows the magnitude of the differences in option price due to varying
q. Choice of q is shown to influence the shape of the implied volatility
smile for varying maturity options.
Many women with a family history of breast and/or ovarian cancer present to
family cancer clinics. The effective counselling and initial management of
these women depends on the estimation of two key related risks: i) the risk
that a mutation in one of the known high penetrance breast ovarian cnacer
susceptibility genes is segregating in the family, and ii) the breast and
ovarian cancer risks to the individual (depends on (i)). Several model have
been developed to estimate these risks and are currently used in clinical
practice. However, each of these models has drawbacks. In particular, no
model currently in use allows for the fact that other breast cancer
susceptibility genes may exist.
We used data from both a population based series of breast cancer cases and
high risk families in the UK, with information on BRCA1 and BRCA2 mutation
status, to investigate the genetic models that can best explain familial
breast cancer outside BRCA1 and BRCA2 families. We also evaluated the
evidence for risk modifiers in BRCA1 and BRCA2 carriers. We estimated the
simultaneous effects of BRCA1, BRCA2, a third hypothetical gene 'BRCA3', and
a polygenic effect using segregation analysis. The hypergeometric polygenic
model was used to approximate polygenic inheritance and the effect of risk
modifiers.
BRCA1 and BRCA2 could not explain all the observed familial clustering. The
best fitting model for the residual familial breast cancer was the
polygenic, although a model with a single recessive allele produced a
similar fit. There was also significant evidence for a modifying effect of
other genes on the risks of breast cancer in BRCA1 and BRCA2 mutation
carriers. Under this model, the frequency of BRCA1 was estimated to be
0.051% (95% CI: 0.021 - 0.125%) and of BRCA2 0.068% (95% CI: 0.033 -
0.141%). The breast cancer risk by age 70 years, based on the average
incidence over all modifiers was estimated to be 35.3% for BRCA1 and 50.3%
for BRCA2. The corresponding ovarian cancer risks were 25.9% for BRCA1 and
9.1% for BRCA2.
I will discuss the implications of our model for genetic counselling and the
potential for its further development.
We consider the single server queue with service in random order. For
a large class of heavy-tailed service time distributions, we determine
the asymptotic behavior of the waiting time distribution. For the
special case of Poisson arrivals and regularly varying service time
distribution with index -\nu, it is shown that the waiting time
distribution is also regularly varying, with index 1-\nu, and the
pre-factor is determined explicitly.
Another contribution of the paper is the heavy-traffic analysis of the
waiting time distribution in the M/G/1 case. We consider not only
the case of finite service time variance, but also the case of
regularly varying service time distribution with infinite variance.
This talk will describe recent work on the use of percolation-based
approaches to represent the spread of fungal diseases in agricultural
crops. In particular it will focus on methods for fitting these models
to experimental observations in a Bayesian framework. Markov chain
methods for investigating parameter posterior densities will be
formulated
and illustrated by application to data from a joint project with
Cambridge University
Stylised facts for univariate high-frequency data in finance are
well-known. They include scaling behaviour, volatility clustering, heavy
tails, and seasonalities. The multivariate problem, however, has
scarcely been addressed up to now. In this work, bivariate series of
high-frequency FX spot data for major FX markets are investigated. First,
as an indispensable prerequisite for further analysis, the problem of
simultaneous deseasonalisation of high-frequency data is addressed. In the
bulk of the study we analyse in detail the dependence structure as a
function of the time scale. Particular emphasis is put on the tail
behaviour, which is investigated by means of copulas and spectral
measures.
(joint work with W. Breymann and P. Embrechts, ETH)
Multivariate extreme value theory and methods concern the
characterisation, estimation and extrapolation of the joint tail of
the distribution of a d-dimensional random variable. Existing
approaches are based on limiting arguments in which all components of
the variable become large at the same rate. This limit approach is
inappropriate when the extreme values of all the variables are
unlikely to occur together or when interest is in regions of the
support of the joint distribution where only a subset of components
are extreme. Motivated by the asymptotic form of the joint
distribution of a d-dimensional random variable conditional on it
having an extreme component, we develop an entirely new
semi-parametric approach which overcomes these existing restrictions
and can be applied to problems of any dimension. The performance of
our approach is demonstrated on simulated and environmental data, and
the new approach is found to compare favourably with existing methods.
We consider the following model of information spreading
(called frog model): there are active and sleeping particles which
live in a d-dimensional lattice. Active particles perform simple
random walks independently of everything, and sleeping particles
become active when hit by active particles. For this model, we prove
a shape theorem, and get some phase transition results.
We consider the choices available to a defined contribution (DC) pension
plan member at the time of retirement for conversion of his pension fund
into a stream of income in retirement. In particular, we compare the
purchase at retirement age from a life office of a conventional life annuity
(that is, a bond-based investment) with distribution programmes that involve
differing exposures to equities during retirement. The residual fund at the
time of the plan member's death can either be bequested to his estate or, in
exchange for the payment of survival credits while alive, reverts to the
life office.
Consider a graph with $n$ nodes numbered $1,\ldots , n$.
For any $i < j $, a {\it link} $i \to j$
exists with probability $p \in (0,1)$
independently of everything else. If it exists, its {\it length} is
$1$.
For $i_1 < i_2 < \ldots < i_l$, a {\it path} $i_1 \to i_2 \to
\ldots \to i_l$ exists (and has a {\length} $l-1$)
if all links $i_j \to i_{j+1}$ exist.
Denote by $L_n$ the maximal path length. We study the asymptotic
properties of $L_n$ when $n$ tends to $\infty$ as well as a
number of related problems. We prove SLLN, functional SLLN,
CLT, functional CLT; estimate the parameters; establish the
Perfect Simulation Algorithm.
Possible generalizations and
applications of the model will be discussed too.
This work has arisen out of the genISYS project, a collaboration
between the University of Edinburgh and Heriot-Watt University. The
Image Systems Engineering Laboratory (ISEL) at Heriot-Watt takes a
broad approach to computer modelling that includes visualisation and
use of modelling formalisms.
Models of biological systems are widely used in artificial
intelligence, optimisation and other areas of computing. Some of
these models, such as artificial neural networks, genetic algorithms,
genetic programming and artificial immune systems, are described and
their applications outlined. Work on specific applications of
evolutionary computing techniques to optimisation problems in cancer
genetic risk analysis, being done within ISEL, is described. Some of
the problems faced and possible solutions being worked on are
outlined.
This work builds on collaborations between biologists and computer
scientists, which has the potential to lead to an extremely useful
cross-fertilisation of ideas and skills between the disciplines. One
of the aims for the future is to prospectively influence the
collection of data, ensuring it is in formats that facilitate the use
of new techniques, such as biological models, to explore innovative
approaches to clinical and epidemiological problems such as cancer
genetic risk analysis.
In this talk, we will present an overview of the Skorokhod reflection
problem on the positive orthant, applied to a special class
of Levy processes. We will study some structural properties
of the "single-class-type" reflection mapping, and then
apply it to the show existence of a stationary distribution
for a reflected Levy process, under a natural stability
condition. We will also look at some structural properties
of the stationary distribution (bounds on tails and
existence of product form).
Finally, we will discuss some open (challenging) problems
We begin with a brief overview of events affecting
with-profit policies and insurers' free assets in recent
years. We then show how the guarantees inherent in
unitised with-profit policies can be priced using options.
We then consider the case where the insurer makes charges
for the guarantees equal to the cost of the matching options.
However, instead of buying the options, the charge is passed
to a guarantee account. The guarantee account is invested in
cash.
At maturity the guarantee account is used to make up any
shortfall between the value of the policyholder's fund and
the guarantee.
Finally, we show how the guarantee fund builds up over a 50
year period for an insurer open to new business. Results are
produced using a Wilkie model to simulate the returns on the
assets in the policyholders fund and the guarantee account.
Of particular interest is the proportion of simulations in
which the guarantee fund becomes exhausted.
We study the utility indifference approach for the valuation
and hedging of contingent claims which integrate tradable and untradable
sources of financial risk.
Such a valuation basically inherits all `desirable properties' of the
classical (static) exponential premium principle, is consistent with
no-arbirage theory, and moreover constitutes a convex measure of risk.
Constructive results are obtained in two classes of models for
tradable/nontradable sources of risk: a multiperiod semi-complete product
setting and a Cox-Ito model.
The split capital investment trust industry is being investigated by the
Financial Services Authority and is the subject of a Treasury Select
Committee enquiry. This seminar will discuss the underlying reasons
for the splits crisis.
We consider asymptotically time- and space-homogeneous Markov chain X_n that
takes values on the real line and has increments possessing a finite
exponential moment. Asymptotically homogeneous Markov chains appear, for
instance, in the perturbed queueing FCFS model or perturbed risk model. The
asymptotic behaviour of the probability P{X_n>x} is studied as n, x\to\infty.
In particular, we extract the ranges of n within which this probability is
asymptotically equivalent to the tail of stationary distribution.
The main tool for our study is the Cramer transform over the distribution of
the chain. The problem is that, being transformed, the chain is no longer the
probabilistic object, in general. We call the new object Markov evolution of
masses. The increments of Markov evolution of masses may have
total mass greater or less then 1. We provide some theory for Markov
evolution of masses such as the analoque of CLL.
We consider a random walk with i.i.d. increments S_n=X_1+...X_n
such that M=sup S_n is finite a.s.
It is well-known that if the mean EX_1 exists, then M is finite a.s.
if and only if EX_1 <0. In the case when EX_1 does not exist, the
conditions for the finiteness of M are also known (Eriksson, 1973).
We are interested in the asymptotics for P(M>x), x\to \infty, when
X_i have a heavy-tailed distribution.
In the finite mean case, the desired asymptotics were obtained
by Veraverbeke (1977).
We present our new results in the ``infinite mean'' case E|X_1|=\infty.
Eilers & Marx (Stat. Sci., 1996) introduced P-splines. There
are two ideas: use B-splines as the basis for the regression and use a
difference penalty to smooth the regression coefficients. We describe
how this method can be used to smooth 2-dimensional mortality data and
how the method leads naturally to the projection of mortality rates.
Properties of the projection are discussed and the role of the order of
the penalty is examined. The methods are illustrated using a large data
set with ages 11 to 100 and years 1947 to 1999.
This talk is devoted to a study of the integral of the workload
process of the single server queue, in particular during one busy
period. Firstly, we find asymptotics of the area A swept
under the workload process W(t)during the busy period when the
service time distribution has a regularly varying tail. We also
investigate the case of a light-tailed service time distribution.
Secondly, we consider the problem of obtaining an explicit
expression for the distribution of A. In the general
GI|G|1 case, we use a sequential approximation to find the
Laplace-Stieltjes transform of A. In the M|M|1 case, this
transform is obtained explicitly in terms of Whittaker functions.
Thirdly, we consider moments of A in the GI|G|1 queue.
Finally, we show asymptotic normality of the integral of the workload
process.
Important features of survival data concerning onset of
inherited disorders are (i) penetrance of the mutation causing the
disorder, meaning that presence of the mutation need not confer 100\%
risk of suffering the disorder; (ii) ascertainment bias, arising because
the families studied are often those with unusually severe histories of
the disorder; and censoring which is present as usual. Gui & Macdonald
(2002) suggested a Nelson-Aalen estimate for a certain function of
the rate of onset; here we study how this function is affected by the
penetrance of the disorder, the censoring and ascertainment bias that
might be present in the sample. The model is applied to the Early-onset
Alzheimer's disease using the PSEN-1 data given in Gui & Macdonald
(2002). We obtain a family of transition intensities, each depending
on the product of the sample penetrance and the probability of being a
mutation carrier given that they are observed.
The talk will discuss an analysis of the impact of credit risk (using the
Jarrow-Lando-Turnbull model) and mortality risk (using a simple stochastic
model of mortality) on the ongoing management/solvency of an annuity book.
I am going to suggest a detailed analysis of basic
notions and results in probability and statistics.
Specific statements or constructions, called
counterexamples, help us to understanding
better the conditions under which important
results are true or false. Thus, the counterexamples
are also results, but knowing and using them
systematically can eventually prevent
us from falling in "traps" of which our "roads" are full.
Curious and unexpected facts will be presented
for popular distributions. Several open questions
will be explicitly outlined.
How to find a good approximation on
the stop-loss premiums of the first and second
orders in the individual risk model with
independent claims occurrences?
How to measure the impact of dependence
between claims occurrences?
How to derive extremal properties of Rademacher functions
or find the best constants in the Rosenthal inequality?
Stochastic orderings and operator inequalities are
employed to answer these and similar questions.
We review and extend
results related to optimal scaling of Metropolis-Hastings algorithms. We
present various theoretical results for the high-dimensional limit.
We also present simulation studies which confirm the theoretical results
in finite dimensional contexts.
The work presented has arisen out of the genISYS project, a collaboration
between the Clinical Genetics and General Practice Departments of the
University of Edinburgh and the Image Systems Engineering Laboratory (ISEL)
at Heriot-Watt University. ISEL takes a broad approach to computer
modelling, that includes both visualisation and use of modelling formalisms.
Models of biological systems are widely used in artificial intelligence,
optimisation and other areas of computing and some of the commoner models
and their applications are of potential use in cancer genetics. ISEL is
working on specific applications of evolutionary computing techniques to
optimisation problems in cancer genetic risk analysis.
There are a number of methods of cancer genetic risk analysis currently
used in the clinical situation and there are important issues surrounding
the computerisation of this process. In particular, the incorporation of
epidemiological data in cancer genetic risk analysis methods applied to
individuals is discussed, and the potential use of evolutionary computing
optimisation methods in this area described.
Although the bulk of income distribution in most economies
follows a log-normal distribution,
there is invariably a power-law tail for the rich, as was
first noted by Pareto and rediscovered subsequently by various authors.
Similar observations hold for the turnovers of businesses.
Various plausible models for economic exchanges can lead to both
log-normal and power law wealth distributions, as we outline.
In the talk we concentrate in particular on the fat cats, taking a Pareto
power-law distribution of individual wealths as given and discuss the
effects of varying the parameters in such a model (i.e. by increasing
taxes) when there is a finite total wealth for the entire economy.
The work caught the eye of the Nature web site because
of a "wealth condensation" which appears for suitable parameter values.
This turns out to be very similar in nature to processes
seen in various classical Urn, i.e. "balls in boxes", models and to a
model of axon growth which Kostya and Reya Khanin have discussed
recently.
Finite model theory is all about what one can say about
classes of finite structures (such as graphs, strings and so on) using
logic; and computational complexity is all about what one can compute on
finite inputs within given resources. There is a very strong link
between finite model theory and computational complexity theory
(exemplified by Fagin's Theorem that a problem is in NP if and only if
it can be defined in existential second-order logic). Often, this link
is strongest when the finite structures are (essentially) strings: on
arbitrary finite structures, the link between resource-bounded
computation and logical definability is nowhere near as clear-cut. In
this introductory talk, I will introduce this subject, known as
descriptive complexity, and I will also introduce models of computation,
program schemes, for computing on arbitrary finite structures, and show
how a consideration of these models can lead to new results in finite
model theory and descriptive complexity. The talk will be introductory
in nature and suitable for a general audience.
The speaker is Co-ordinator of MathFIT. The broad aim of the Mathematics
for Information Technology initiative is to support, through research
grants, visiting fellowships, networks, workshops and summer schools,
high-quality interdisciplinary research in areas at the interface
between mathematics and computer science. It is jointly sponsored by the
Engineering and Physical Sciences Research Council (EPSRC) and the
London Mathematical Society (LMS), and began in the summer of 1996, was
subsequently expanded in spring 2000 and will run until 2003. Following
his research talk (ca. 45 min), the speaker will give on overview of
MathFIT (ca. 15 min).
The Greek astronomer, geographer and mathematician, Claudius
Ptolemy, included near the start of his great work on astronomy, "The
Almagest", 13 pages (out of 650 in translation) on how to construct a table
of Chords of angles.
Note that: Chord(x) = 2 sin(x/2).
Ptolemy used only Euclidean-style geometry, and yet calculates in effect a
table of sines of angles at 1/4 degree intervals accurate to six decimal
places. And this in spite of the appalling Greek numerical notation! It is
brilliant piece of practical mathematics, and predates conventional
trigonometry in India by some hundreds of years.
New criteria for instability of Markov chains will be presented.
Then the results will be applied to the study of a random walk
in the case when the drift does not exist.
Also, a number of examples will be considered.
Professor Robert Brown will report on two recent pieces of research
on which he has been working. Both have to do with the Macro-economic
impacts of Population Aging on Retirement systems.In the first paper,
he will argue that it is inevitable that the labour force retirement age
will rise between sometime after 2006. Depending on the level of labour
force productivity that we can achieve, this rise in the retirement age
may not have a large political impact. However, once incentives for early
retirement change into incentives for later retirement, we can expect
some kind of behaviourial response from the work force. In particular,
we should expect demands for more flexible retirement systems as workers
attempt to smoothly transit into retirement. Many of the requests for
pension flexibility are now obviated by pension legislation. Thus,
this legislation will have to be questioned and (hopefully)
redesigned.
In the second paper, Professor Brown will argue that our present system
of Registered Pension Plans (RPPs) and Registered Retirement Savings
Plans (RRSPs) will provide the government(s) with exactly the correct
amount of cash flow and at exactly the right time, to pay for the
increased demand for health care created by the aging baby boomers.
Thus, accidentally, we may have created the perfect macro-economic
immune portfolio (i.e. RPP/RRSPs versus Health Care costs). However,
this is dependent upon the government not looking at RPPs/RRSPs as a
source of Tax Expenditures but rather as the perfect deferred tax asset.
In particular, the government must embrace a philosophy whereby the
RPP/RRSP system will be allowed to expand as rapidly as per unit health
care costs are allowed to rise.
We study distributions F on positive half line such that for some positive T,
the second convolution F^{*2}(x,x+T] is equivalent to 2F(x,x+T] as x tends to
infinity. The case of infinite T corresponds to F being subexponential, and
our analysis shows that the properties for finite T are, in fact, very
similar to this classical case. A parallel theory will be presented in the
presence of densities. Applications are given to random walks, the key
renewal theorem, compound Poisson process and Bellman-Harris branching
processes.
Joint work with Soeren Asmussen and Serguei Foss.
In Random Sequential Adsorption, each successive particle
is deposited uniformly at random onto a d-dimensional
region, subject to non-overlap with predecessors. Chemists
and others have made numerous simulation studies
of such models for d=2, but rigorous theory (the Renyi car-parking
model) has been limited to d=1. We describe recent
work trying to redress this imbalance.
This work investigates capital adequacy risk measures for
insurance asset portfolios focusing on coherence and second order
stochastic dominance. Risk measures currently used for insurance
regulation will be examined and generalized to the class of distortion
risk measures. Necessary and sufficient conditions for coherence and
for strict consistency with second order stochastic dominance are
illustrated.
The public policy debate about the use of genetic information in the
formation of insurance contracts in the United Kingdom has at times
been influenced as much by rhetorical argument as by the findings of
actuarial research. Issues such as the scope for genetic
discrimination and the low uptake of genetic tests have been presented
as potentially serious social problems with little, if any,
supporting evidence. In a recent study conducted for the Association
of British Insurers I attempted inter alia to map the state of
actuarial research in relation to genetics and insurance. This study
examined some of the more contentious issues in the public policy
debate against the existing research base. This work revealed
significant gaps in the state of knowledge about the potential impact
of genetic information on a range of insurance products and indicated
a possible future research agenda. It also raised interesting
questions about the role, function and interpretation of objective
actuarial research in a contested public policy debate.
Biography : Dr Tony McGleenan
Dr Tony McGleenan graduated in Law from Queen's University. He
commenced his academic career as a Teaching Fellow in the School of
Law in 1992. In January 1994 he was appointed to the post of
Lecturer in Jurisprudence at Queen's and became Senior Lecturer in Law
in 2000. He studied at the Honourable Society of King's Inns in Dublin
was called to the Bar of the Republic Of Ireland. In October 1997 he
was also called to the Bar of Northern Ireland. His doctoral research
was in the legal and ethical implications of advances in genetic
technology.
In 1999 he was winner of the first Queen's University Teaching Award
for the integration of information technology in undergraduate
teaching.
His main research interests are in the area of medical law and
biotechnology.
He has published widely in legal and medical journals and is the
co-author of Genetics and Insurance (1999). He is President of the
Northern Ireland Forum for Healthcare Ethics and Law and has acted as
consultant and advisor to a wide range of bodies including the
Association of British Insurers and the Science and Technology Options
Assessment Unit of the European Parliament. He was appointed to
Northern Ireland Human Organs Inquiry in 2001 and also sits on the
Clinical Ethics Committee of the Royal Group of Hospitals and the
Council of the Pharmaceutical Society of Northern Ireland. He
maintains a private legal practice at the Bar of Northern Ireland
specialising the field of Public Law.
Spectrally positive Levy processes are processes with
stationary independent increments and positive only jumps.
Excluding compound Poisson processes, they often appear
in approximations of stochastic systems in several applications,
notably in Internet traffic with heavy-tailed session durations.
Such traffic goes through several nodes and bottlenecks
and it is frequently desirable to obtain information on the
occurrence of rare events such as the event that the queue
at a particular node exceeds a certain threshold.
In this talk, we study events of the following types:
of a large terminal value and occurrence of a large maximum
within a period of time, both for the Levy process itself
and also for the reflection of the Levy process with negative drift.
We prove limit theorems that describe the way that various
events occur, using the explicit representation of the Levy
process in terms of a 2-dimensional Poisson random measure.
As a by-product of our techniques we obtain a new proof
for the asymptotic behavior of the tail of the stationary
distribution for the reflected process.
In the past decade, two techniques have been proposed to deal
with variable dimension models within the Bayesian framework,
namely the Reversible Jump MCMC technique of Green (1995) and
the Birth-and-Death process of Stephens (2000). In this talk,
we provide introduction to both methods and show how close they
are to one another.
Ref. http://www.ceremade.dauphine.fr/~xian/ctrjmcmc.ps.gz
This talk is devoted to the stability of the ruin probability with respect
to the parameters governing the risk process (such as inter-occurrence
times distribution, claim size distribution, etc.). We consider a risk
model with Levy processes driven investments. To derive quantitative
stability bounds for the probability of ruin we use Markov chains and
regenerative processes approaches. The first approach provides bounds in
the weighted total variation distance while the second one leads to bounds
in the weighted uniform distance which is more suitable for practical
applications.
Multi-class systems are of increasing importance in the
practical modelling world but present a significant challenge for
analysis. Most results to date concerning the optimal dynamic
control of service in such systems have assumed (holding) costs to
be incurred at rates which are linear in the number of customers
present. In response to arguments that such an assumption is
often inappropriate, we develop a simple index heuristic for a multi-
class M/M/1 system with increasing convex holding cost rates. We
use a prescription of Whittle's as the basis of the development of
the required indices. A numerical study elucidates very strong
performance of the index policy. Note that most of the ideas and
results also hold for multi-class M/G/1 systems under additional
conditions. However, the analysis of this case is considerably more
difficult and will not be covered in the talk.
(Work is joint with P.S.Ansell and M.O'Keeffe of Newcastle
University and J.Nino-Mora of Universitat Pompeu Fabra,
Barcelona).
We discuss the valuation problem for claims in case the price process of
some risky asset is influenced by some exogenous random factor. Typically
this factor is modelled by some unbounded stochastic process, and this can
lead in some approaches to the occurence of singularities. It turns out
that typically it is still possible to follow an entropy approach. We
calculate the density of the resulting pricing measure for some classical
stochastic volatility models and provide the relevant verification
results. Moreover, we discuss the structure of the filtration generated by
the price process.
An approach, that is an extension of the Hull & White model (1987), is
employed for pricing European options under the assumption of a mean
reverting volatility for the underlying asset. The approach uses a Taylor
series expansion method to approximate the price of a European call option
in a market with no arbitrage opportunities. The transition to a
risk-neutral economy is accomplished by introducing an equivalent martingale
measure based on the findings of Romano and Touzi (1997). Numerical results
are obtained and compared with similar studies Lewis (2000).
Genetic testing is a concern for insurers if they cannot use test results
in underwriting. We model adverse selection in an insurance market with
genetic testing for breast and ovarian cancer. Increased forces of
mortality resulting from a family history of cancer or a positive test
for a BRCA mutation are calculated. Using a Markov model, we estimate
costs of adverse selection, assuming various testing and insurance
purchase behaviors. Previous results are extended by introducing an
elastic demand function in the analysis.
We propose a structure for modelling fixed-income bond prices with a
view to application in the management of long-term interest-rate risk.
The model exploits the framework developed by Flesaker &
Hughston (1996) which provides a straightforward means of ensuring
that nominal rates of interest remain positive.
The structure of the model allows us to include some factors which
ensure realistic modelling of short-term dynamics while other factors
have a much longer-term impact. The model produces two useful effects:
sustained periods of both high and low interest rates; a wide range
of par yields on long-dated or irredeemable bonds. These effects
coexist comfortably with economically reasonable short-term dynamics.
Finally we note that the pricing of European-type derivatives is
straightforward.
For a stable single-server GI/GI queue, the following is well-known:
for k>0, the stationary waiting time has a k-th finite moment iff
the service time has a (k+1)-st finite moment. However, only certain
sufficient conditions for existence of moments for stationary waiting
time are known in the multi-dimensional case.
We formulate new results in this direction. In particular, in the
so-called maximal stability case, the necessary and sufficient
conditions will be formulated. Also, under certain conditions,
the tail asymptotics has been found.
The value function of an American put option defined in a discrete domain
may be given as a solution of a Linear Complementarity
Problem (LCP). However, the state of the art methods that solve LCP
converge slowly. Recently, Dempster, Hutton & Richards have
proposed a Linear Program (LP) formulation of the American put and a
special simplex algorithm that exploits the option structure. They
give numerical examples with run times which grow almost linearly with the
number of spatial grid points. Based on these ideas we show
in a constructive fashion that a new algorithm may be devised which
processes the original LCP in linear number of spatial grid points.
The Lindley queueing theory recursion, or workload process, is, under
appropriate conditions, related to a random walk modulated by a
regenerative process. In particular the stationary distribution of the
workload process coincides with that of the maximum of the random walk.
We derive the asymptotics of this distribution under the assumption that
the increments of the modulated random walk are heavy-tailed. The results
generalise the now classical theorem of Veraverbeke for unmodulated random
walks.
We compare various approaches to the modelling of categorical animal
behaviour data. In particular, we consider a large dataset of binary
feeding data and look at methods of fitting hidden Markov models,
semi-Markov models and latent Gaussian variable models. We discuss
methods of parameter estimation and consider the ease with which the
models can be extended to incorporate additional information such as
diurnal cycles. We then go on to discuss techniques of model
comparison, advocating a parametric bootstrap approach which can be
used to assess the relative fit of the three types of model. The
models are non-nested and fit according to different criteria,
nevertheless this simulation-based method is straightforward to apply.
We present results of the ongoing research "Genetic knowledge of
APKD and its implications for Critical Illness Insurance".
Results are presented for three stages: a) Modelling of intensities;
b)Differences in premiums; c) Effect of moratoria and adverse selection.
In this talk I will describe a problem with the following characteristics:
- the risk-free rate of interest is stochastic
- additionally there are risky bonds and other risky assets
- the investor has a pensions contract invested in these assets
- he pays premiums at a constant percentage of salary, p.S(t)
- salary, S(t), is stochastic and incorporates non-hedgeable risks
- at retirement the investment fund, F(T), is used to purchase an annuity
i.e. P(T)=F(T)/a(T)
- the annuity price, a(T), depends upon the market rates of interest at
retirement
The investor has a terminal utility function which is a function of
pension as a proportion of final salary i.e. U(P(T)/S(T).
The problem is how should he invest in order to maximise his expected
terminal utility.
I will discuss the (limited) progress to date and outstanding problems.
This talk will discuss several approaches to simplifying stochastic
spatial models by using deterministic systems of ODEs to represent the
evolution of moments of the process. The use of cluster approximations
will be described and applied to stochastic models for the spread and
control of arboreal viruses. Some improvements to traditional cluster
approximations are proposed the compared in a simulation study. The
talk describes joint research with Dr J Filipe, Dept. Plant Sciences,
University of Cambridge.
We study random walks in a random environment on a regular,
rooted, coloured tree. The asymptotic behaviour of the walks is classified
for ergodicity/recurrence/transience in terms of the geometric properties
of the matrix describing the random environment.The close connection
between various problems on random walks in random environment and the so
called multiplicative chaos martingale will be shown.
I shall give a presentation of the solution to a fusion of two
fundamental problems in mathematical finance. The first problem is that
of maximizing the expected utility of terminal wealth of an investor who
holds a short position in a contingent claim, and the second is that of
maximizing terminal wealth where the utility function allows the
investor to have negative wealth. Under assumptions of reasonable
asymptotic elasticity on the investor's utility function, we can present
an optimal investment theorem and simultaneously treat the corresponding
dual problem.
CDKN2A is the major inherited cause of susceptibility to melanoma
known to date. CDKN2A mutation carriers have an increased risk of
melanoma and families with such mutations have been identified
world-wide. As part of a study being conducted by the Melanoma
Genetics Consortium, a collaboration among geneticists and
oncologists interested in the causes of melanoma, we have estimated
the penetrance (ie the cumulative risk) of melanoma in mutation
carriers. I will discuss the issues of estimating penetrance in
genetic epidemiological studies and show the evidence that for
melanoma this risk varies world-wide.
The talk will describe the development of an actuarial model for
Coronary Heart Disease and Stroke which incorporates the development of
hypertension, hypercholesterolaemia and diabetes.
We analyze the behavior of Generalized Processor Sharing (GPS) queues with
heavy tailed service times. The model consists of coupled queues; each one
receives an arrival stream of customers with inter-arrival time that are
i.i.d and with service times that are subexponential. We calculate the
exact stationary workload asymptotic of an individual flow for this model.
We investigate the valuation of catastrophe insurance derivatives that are
traded at the Chicago Board of Trade. By modelling the underlying index as a
compound Poisson process we give a representation of the set of all
martingales that can be constructed upon that index. This characterisation
enables us to derive closed pricing formulae for every fixed equivalent
martingale measure. Furthermore, we develop clear one-to-one connections
between the set of equivalent measures, the set of no-arbitrage prices, and
the local characteristics of the index. We then compare the results with
actuarial pricing methods. It is shown that premium calculation principles
do not create a one-to-one correspondence between premia and the set of
equivalent probability measures. Following a representative agent approach
we determine the unique equivalent martingale under which prices in the
insurance market are calculated.
Let X and Y be two independent non-negative random variables with common distribution. Assuming they have no exponential moments (in other words, they are heavy-tailed), then the lower limit (as x tends to infinity) of the fraction of P(X+Y > x) over P(X > x) is always equal to 2. This theorem is based, in particular, on results from a paper by Walter Rudin (1973).
We describe a stochastic model of mobility of sensors in R^3
that combines interesting features from stochastic calculus
and stochastic geometry. We call the model
Brownian Boolean Model
as it can be seen as a combination of a Boolean model (based on a Poisson
process) and Brownian motion. The goal of a sensor network is
to detect a target placed in an unknown position.
We obtain explicit formulas for the
distribution of detection times and discuss ways to bound the
detection probability for different mobility scenaria.
There is an interesting link between the stochastic-geometric capacity
and the classical (Newtonian) capacity of the target set.
We study the optimal portfolio choices of an investor in an illiquid
market where all trading takes place in discrete time. The liquidity
structure assumed are in the spirit of recent works by Cetin, Jarrow and
Protter (2004) and Rogers and Singh (2004). The investor aims to
maximize her utility from terminal wealth. We show that the marginal
utility of the optimal wealth serves as the Radon-Nikodym derivative to
turn the marginal price process defined by the optimal strategy a
martingale. The link between the optimal strategy and absence of
arbitrage is discussed and a numerical study of an investor with CARA
utility and a position in
a put option is given in a binomial framework.
In contrast to the classical case of frictionless market, in
models financial markets with transaction costs (even in discrete time)
there is a variety of natural definitions of no-arbitrage properties.
In the latter theory the concept of martingale density is replaced
by a concept of martingale moving in a dual to the solvency region.
Necessary and sufficient conditions for no-arbitrage properties for
various definitions will be discussed.
Coupling is a beautiful and elegant technique for
investigation of probabilistic questions, with applications including
epidemic theory and particle systems, through perfect simulation, to
probabilistic approaches to differential equations. We will focus on a
classic coupling problem which lies at the heart of many applications:
given a random process, can one construct two coupled realizations
starting from different initial points and yet meeting almost surely?
And if so, then can one do so without "cheating" - using future
information about one of the processes to build the other? And if so,
then can one add requirements to couple further path-functionals of the
process - to what degree can one achieve /exotic/ coupling? In this talk
I will give a general survey of the area and describe some of my recent
work on exotic coupling.
Over the past decades or so, there has been considerable interest on developing accurate and appropriate methods for risk measurement and management. Various measures for risk have been introduced and investigated theoretically and empirically. Value at Risk (VaR) has emerged as a popular tool for risk measures and become the workhorse of risk management practice. In this talk, I will first provide a snapshot for some contemporary research on risk measures, such as VaR and coherent risk measures. The risk behavior of linear portfolios has been investigated extensively and well documented in the finance and insurance literature. Recently, the spotlight has turned to developing theoretically consistent and practically useful quantitative techniques for measuring and managing risk for non-linear instruments, such as options. In the second part of my talk, I will present some research work on quantitative risk measures for derivative instruments and suggest some potential topics for further investigation.
We look at queues formed by traffic and congestion in communications
networks. We are interested in models in which ``flows'' or ``calls'' have
simultaneous capacity requirements from a number of resources, each of
which can share capacity over all ``flows'' present.
We consider the problem of stability. Namely under
which control strategies is our system stable. The system is stable
if the number of ``flows'' does not increase to infinity.
Clearly it is necessary for the capacity of each resource to be
greater than the total service per unit time required from that
resource. It might be thought that this condition would also be
sufficient
for stability in ``non-idling'' control strategies. However this
turns not to be the case. This is essentially due to the phenomenon
called entrainment, whereby the capacity required by flows of a
given type is indeed available at each resource but at different
times. This causes a violation of the simultaneity requirements of the
system.
We will use Lyapunov functions to prove our main results.
The minimal entropy and minimal martingale measures are shown to be
related by an Esscher transform, involving the mean-variance trade-off, in
an incomplete diffusion model containing a traded stock and a non-traded
stochastic factor. The coefficients of the diffusions are adapted to the
Brownian motion driving the non-traded factor, as is typical in stochastic
volatility models. The result is motivated by a formal analysis of
exponential indifference prices, and made rigorous using a representation
equation for the $q$-optimal measure due to Hobson. The analysis yields a
new representation for the marginal price of a claim on the non-traded
factor. Specialising to a lognormal model, we derive explicit formulae for
indifference prices and hedging strategies for the claim, using power
series approximations. These are used to conduct a simulation study of
optimal hedging performance.
Phylogenetic trees, also known as evolutionary trees,
model the evolution of biological species or genes
from a common ancestor. Most computational problems
associated with phylogenetic tree reconstruction are
very hard (specifically, they are NP-hard, and are
practically hard, as real datasets can take years
of analysis, without provably optimal solutions
being found). Finding ways of speeding up the
solutions to these problems is of major importance
to systematic biologists. Other approaches take only
polynomial time and have provable performance guarantees
under Markov models of evolution; however, our recent
work shows that the sequence lengths that suffice for
these methods to be accurate with high probability grows
exponentially in the diameter of the underlying tree.
In this talk, we will describe new dataset decomposition
techniques, called the Disk-Covering Methods, for
phylogenetic tree reconstruction. This basic algorithmic
technique uses interesting graph theory, and can be used to
reduce the sequence length requirement of polynomial
time methods, so that polynomial length sequences
suffice for accuracy with high probability (instead of
exponential). We also use this technique to speed up
the solution of NP-hard optimization problems,
such as maximum likelihood and maximum parsimony.
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Slides for this talk
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This talk will outline some potential applications of Markov chain
methods to philosophies of statistical inference other than the Bayesian
one. It will describe how the applicability of approaches such as
Fisher's fiducial inference could be extended using Markov chain
methods, and will attempt to define some potential research problems
associated with Markov chain that arise in the process. Volunteers for
collaboration on a research proposal may be sought.
If X is a sequence space with a finite state space,
then a g-measure is a shift-invariant probability measure on X
with certain conditional probabilities specified by a function g.
We will discuss the question of whether or not such measures are
uniquely determined by g, focusing particularly on some recent
results.
Data with an array structure are common in statistics.
An early example is the factorial design and Yates (1937) gave an
efficient algorithm for computing the factorial effects in such a design.
The generalized linear model (GLM) of Nelder & Wedderburn (1972) gives a
unified approach to analysing regression problems with non-normal error
structure. However, this analysis ignores any array structure in the
data or the model. We develop an arithmetic of arrays which generalizes
Yates algorithm and which allows us to define the expectation of a
data array as a sequence of linear operations on a coefficient array.
This arithmetic also leads to low storage, high speed computation in
the scoring algorithm of the GLM. We call such a model a generalized
linear array model (GLAM). We apply the method to the smoothing of
multidimensional arrays.
In this paper we study distances and connectivity properties of random graphs with
an arbitrary i.i.d. degree sequence. When the tail of the degree distribution
is regularly varying with exponent 1-τ there are three distinct cases:
(i) τ > 3, where the degrees have finite variance,
(ii) τ ∈ (2, 3), where the degrees have infinite variance,
but finite mean, and
(iii) τ ∈ (1, 2), where the degrees have infinite mean.
These random graphs can serve as models for complex networks where degree power
laws are observed. The distances between pairs of nodes in the three cases
mentioned above have been studied in three previous publications, and we
survey the results obtained there. Apart from the critical cases
τ = 1, τ = 2 and τ = 3, this completes the scaling picture.
We explain the results heuristically and describe related work and open problems.
We also compare the behavior in this model to Internet data, where a degree
power law with exponent τ ≈ 2.2 is observed. Furthermore,
in this paper we derive results concerning the connected components
and the diameter. We give a criterion when there exists a unique largest
connected component of size proportional to the size of the graph,
and study sizes of the other connected components.
Also, we show that for τ ∈ (2, 3), which is most often observed
in real networks, the diameter in this model grows much faster than the
typical distance between two arbitrary nodes.
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Paper by
Remco van der Hofstad, Gerard Hooghiemstra, and Dmitri Znamenskiz
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We propose a new dynamic hedging strategy based on the tools
of fractional calculus. We compare the profit and loss (P&L)
resulting from hedging vanilla options when the classical approach
of Delta- and Gamma-neutrality is employed, to the results
delivered by what we label Delta- and Fractional-Gamma-hedging.
For specific cases, such as the FMLS of Carr and Wu (2003) and
Merton's Jump-Diffusion model, the volatility of the P&L is
considerably lower (in some cases only 25%) than that resulting
from Delta- and Gamma-neutrality. We also show that the pricing
equation satisfied by European-style options, written on
securities that follow some of the most widely used jump
processes, satisfy a fractional PDE.
In my talk I will discuss the application of
probabilistic modelling to three problems
in computational molecular biology:
the prediction of mosaic structures in
DNA sequence alignments, the reverse engineering
of local gene regulatory networks from
transcriptomic data, and the in silico
prediction of protein interactions
from primary sequences.
This talk is concerned with a combinatorial lemma that is
central to the proof of the Poisson hypothesis for large
queueing networks.
Now that the human genome has been sequenced, emphasis is shifting
toward the identification and characterization of all the functional
elements in the genome. This work is crucial for identifying the causal
variants that confer inherited susceptibility to complex disease, as
well as variable sensitivity to drugs and other environmental factors.
One approach is to identify functional elements based on the presence
of variability that results in expression level differences. Such
expression levels are often monitored using microarrays, typically
either two-color spotted arrays or oligo arrays such as those from
Affymetrix. The low-level analysis of such arrays has provided some
challenging statistical problems. In this talk I will discuss the
low-level analysis of data obtained from the Illumina expression
platform, which is based on a bead technology. I will also discuss some
of the statistical issues that arise in the analysis of whole-genome
scans that attempt to correlate expression levels with single
nucleotide polymorphisms.
We review several general results concerning indifference
pricing in two-factor Markovian models, with an emphasis in the
abstract duality theory underlying them, and then focus on exactly
solvable models for pricing European-style volatility derivatives.
Turning to American-style derivatives, we conclude with a discrete-
time algorithm for pricing real options in the absence of a perfectly
correlated asset, in particular for finding optimal exercise policies
for executive options.
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We consider a classical risk model modified by the
introduction of dividends. We assume that when the insurer's surplus is
above a specified level b (the dividend barrier), part of the premium
income is paid to shareholders as dividends. We derive a general
expression for the expected present value of dividend income to
shareholders. In the special case when individual claim amounts are
exponentially distributed, we consider the question of finding the
optimal combination of dividend barrier and dividend rate subject a
constraint on the insurer's ruin probability.
This seminar forms a preliminary presentation of the material of my PhD
thesis. We study stochastic queueing networks which are described by
Markov chains. A single-server queue and a multi-server queue with i.i.d.
driving sequences, a generalized Jackson network are important examples.
The two questions that we will discuss are existence of moments of main
characteristics of networks and convergence rates of the governing Markov
chains to stationarity. The former will be achieved by studying related
problems for random walks. Applications to continuity theorems for the
stationary workload will be also considered. To achieve the latter goal we
introduce the notion of a monotone separable network with an arbitrary
initial state. The properties of this class of networks allow us to
develop a uniform approach for studying convergence rates. A number of
examples will be given.
This talk will discuss some graphical goodness-of-fit procedures and prediction of distribution functions. The goodness-of-fit
procedures are based on empirical moment generation functions, and the prediction problem is based on kernel smoothing.
We consider M-estimation of a location parameter for processes with
zero autocorrelations but long-range dependence in volatility. The type of central limit theorem depends on the type of the $\psi-$function. In particular surprising is the case where $\psi(-x)=-\psi(x),$ since there, long memory in volatility does not have any effect, even if $\hat{\mu}$ is a nonlinear estimator.
We derive an upper bound for the
largest Lyapunov exponent
of a Markovian random matrix product of nonnegative matrices.
The bound is expressed as the maximum of a nonlinear
concave function over a finite-dimensional
convex set of probability distributions.
The technique used is Markovian type counting.
This is joint work with Reza Gharavi.
This study is intended to gauge the risk inherent in defined contribution
(DC) pension plans on an individual and on an aggregate basis, based on
United States data. Our aim is to gain insight into the consequences of a
DC pension scheme becoming the predominant pillar of retirement income for
an entire society. It is necessary for the primary source of retirement
income to, by design, provide a sufficient pension that will offer
financial security to the elderly and will facilitate the transition from
employment to retirement. Due to the uncertainty in its accumulated
wealth, such a requirement could not be fulfilled by a traditional DC
pension plan if the pension delivery date is fixed. Therefore, rather
than focus on the accumulated wealth at a specified retirement age, this
study investigates the likely retirement age of DC participants if they
hoped to maintain a fixed standard of living once they have retired, which
will sustain them till death. Based on the simulated output of a DC
flexible age of retirement model, we decide upon the optimal investment
strategies. We then examine the demographic dynamics in an entire
population of DC pension plan participants. The conclusions drawn
demonstrate the significant role the market plays in the effectiveness of
the DC pension plan scheme's success or failure. There is a high level of
uncertainty in the age of retirement of each DC participant, regardless of
his or her investment strategy. Furthermore, there are large retirement
age discrepancies between the DC participants in different cohorts,
despite their identical characteristics. We find that, even when we allow
for a wide range of investment strategies amongst the members, the ratio
of retirees to workers varies significantly over time. This suggests that
countries dominated by DC schemes of this type may, over time, be exposed
to significant risk in the size of its labour force.
(The talk is based on a paper by Bonnie-Jeanne MacDonald and Andrew Cairns.)
Seminar Timetable
Serguei Foss / Heriot-Watt University/ foss@ma.hw.ac.uk
