Recursos de colección
Bäuerle, Nicole; Grübel, Rudolf
The classical models in risk theory consider a single type of
claim. In the insurance business, however, several business lines
with separate claim arrival processes appear naturally, and the
individual claim processes may not be independent. We introduce a
new class of models for such situations, where the underlying
counting process is a multivariate continuous-time Markov chain of
pure-birth type and the dependency of the components arises from
the fact that the birth rate for a specific claim type may depend
on the number of claims in the other component processes. Under
certain conditions, we obtain a fluid limit, i.e. a functional law
of large numbers for these processes. We...
Gamarnik, David; Momčilović, Petar
We consider a multiserver queue in the Halfin-Whitt regime: as the
number of servers n grows without a bound, the utilization
approaches 1 from below at the rate Θ(1/√n).
Assuming that the service time distribution is lattice valued with
a finite support, we characterize the limiting scaled stationary
queue length distribution in terms of the stationary distribution
of an explicitly constructed Markov chain. Furthermore, we obtain
an explicit expression for the critical exponent for the moment
generating function of a limiting stationary queue length. This
exponent has a compact representation in terms of three
parameters: the amount of spare capacity and the coefficients of
variation of interarrival and service times. Interestingly,...
Piera, Francisco J.; Mazumdar, Ravi R.; Guillemin, Fabrice M.
In this paper we consider reflected diffusions with positive and
negative jumps, constrained to lie in the nonnegative orthant of
R^{n}. We allow for the drift and diffusion
coefficients, as well as for the directions of reflection, to be
random fields over time and space. We provide a boundary behavior
characterization, generalizing known results in the nonrandom
coefficients and constant directions of the reflection case. In
particular, the regulator processes are related to semimartingale
local times at the boundaries, and they are shown not to charge
the times the process expends at the intersection of boundary
faces. Using the boundary results, we extend the conditions for
product-form distributions in the stationary...
Touboul, Jonathan; Faugeras, Olivier
The problem of finding the probability distribution of the first
hitting time of a double integral process (DIP) such as the
integrated Wiener process (IWP) has been an important and
difficult endeavor in stochastic calculus. It has applications in
many fields of physics (first exit time of a particle in a noisy
force field) or in biology and neuroscience (spike time
distribution of an integrate-and-fire neuron with exponentially
decaying synaptic current). The only results available are an
approximation of the stationary mean crossing time and the
distribution of the first hitting time of the IWP to a constant
boundary. We generalize these results and find an analytical
formula for the first...
Griffiths, Robert C.; Jenkins, Paul A.; Song, Yun S.
The diffusion-generator approximation technique developed by De
Iorio and Griffiths (2004a) is a very useful method of
constructing importance-sampling proposal distributions. Being
based on general mathematical principles, the method can be
applied to various models in population genetics. In this paper we
extend the technique to the neutral coalescent model with
recombination, thus obtaining novel sampling distributions for the
two-locus model. We consider the case with subdivided population
structure, as well as the classic case with only a single
population. In the latter case we also consider the
importance-sampling proposal distributions suggested by Fearnhead
and Donnelly (2001), and show that their two-locus distributions
generally differ from ours. In the case of the
infinitely-many-alleles...
Gentil, Ivan; Rémillard, Bruno
While the convergence properties of many sampling selection
methods can be proven, there is one particular sampling selection
method introduced in Baker (1987), closely related to `systematic
sampling' in statistics, that has been exclusively treated on an
empirical basis. The main motivation of the paper is to start to
study formally its convergence properties, since in practice it is
by far the fastest selection method available. We will show that
convergence results for the systematic sampling selection method
are related to properties of peculiar Markov chains.
Caramellino, Lucia; Pacchiarotti, Barbara
The paper deals with the asymptotic behavior of the bridge of a
Gaussian process conditioned to stay in n fixed points at
n fixed past instants. In particular, functional large
deviation results are stated for small time. Several examples are
considered: integrated or not fractional Brownian motions and
m-fold integrated Brownian motion. As an application, the
asymptotic behavior of the exit probability is studied and used
for the practical purpose of the numerical computation, via Monte
Carlo methods, of the hitting probability up to a given time of
the unpinned process.
Frei, Christoph; Schweizer, Martin
We study the exponential utility indifference valuation of a
contingent claim B in an incomplete market driven by two
Brownian motions. The claim depends on a nontradable asset
stochastically correlated with the traded asset available for
hedging. We use martingale arguments to provide upper and lower
bounds, in terms of bounds on the correlation, for the value
V^{B} of the exponential utility
maximization problem with the claim B as random endowment.
This yields an explicit formula for the indifference value
b of B at any time, even with a fairly general
stochastic correlation. Earlier results with constant correlation
are recovered and extended. The reason why all this works is that,
after a...
Dayanik, Savas; Powell, Warren; Yamazaki, Kazutoshi
A multiarmed bandit problem is studied when the arms are not
always available. The arms are first assumed to be intermittently
available with some state/action-dependent probabilities. It is
proven that no index policy can attain the maximum expected total
discounted reward in every instance of that problem. The Whittle
index policy is derived, and its properties are studied. Then it
is assumed that the arms may break down, but repair is an option
at some cost, and the new Whittle index policy is derived. Both
problems are indexable. The proposed index policies cannot be
dominated by any other index policy over all multiarmed bandit
problems considered here. Whittle indices are...
Charalambous, C.; Gittins, J. C.
Pharmaceutical companies have to face huge risks and enormous
costs of production before they can produce a drug. Efficient
allocation of resources is essential to help in maximizing
profits. Yu and Gittins (2007) described a model and associated
software for determining efficient allocations for a preclinical
research project. This is the starting point for this paper. We
provide explicit optimal policies for the selection of successive
candidate drugs for two restricted versions of the Yu and Gittins
(2007) model. To some extent these policies are likely to be
applicable to the unrestricted model.
Pateiro-López, Beatriz; Rodríguez-Casal, Alberto
The problem of estimating the Minkowski content
L_{0}(G) of a body
G ⊂ R^{d} is considered. For
d = 2, the Minkowski content represents the boundary length
of G. It is assumed that a ball of radius r can roll
inside and outside the boundary of G. We use this shape
restriction to propose a new estimator for
L_{0}(G). This estimator is based on the
information provided by a random sample, taken on a square
containing G, in which we know whether a sample point is in
G or not. We obtain the almost sure convergence rate for
the proposed estimator.
Møller, Jesper; Helisová, Kateřina
We study a flexible class of finite-disc process models with
interaction between the discs. We let U denote the random
set given by the union of discs, and use for the disc process an
exponential family density with the canonical sufficient statistic
depending only on geometric properties of U such as the
area, perimeter, Euler-Poincaré characteristic, and the
number of holes. This includes the quermass-interaction process
and the continuum random-cluster model as special cases. Viewing
our model as a connected component Markov point process, and
thereby establishing local and spatial Markov properties, becomes
useful for handling the problem of edge effects when only U
is observed within a bounded observation window....
Bordenave, Charles; Torrisi, Giovanni Luca
We extend a result due to Zazanis (1992) on the analyticity of the
expectation of suitable functionals of homogeneous Poisson
processes with respect to the intensity of the process. As our
main result, we provide Monte Carlo estimators for the
derivatives. We apply our results to stochastic models which are
of interest in stochastic geometry and insurance.
Casella, Bruno; Roberts, Gareth O.
We describe and implement a novel methodology for Monte Carlo
simulation of one-dimensional killed diffusions. The proposed
estimators represent an unbiased and efficient alternative to
current Monte Carlo estimators based on discretization methods for
the cases when the finite-dimensional distributions of the process
are unknown. For barrier option pricing in finance, we design a
suitable Monte Carlo algorithm both for the single barrier case
and the double barrier case. Results from numerical investigations
are in excellent agreement with the theoretical predictions.
Chakraborty, Santanu; López-Mimbela, Jose Alfredo
We consider a branching particle system where an individual
particle gives birth to a random number of offspring at the place
where it dies. The probability distribution of the number of
offspring is given by
p_{k}, k = 2, 3, .... The
corresponding branching process is related to the semilinear
partial differential equation
∂u / ∂t = Au(t, x) + ∑_{k=2}^{∞}p_{k}(x)u^{k}(t, x)
for x ∈ R^{d}, where A
is the infinitesimal generator of a multiplicative semigroup and
the p_{k}s, k = 2, 3, ..., are
nonnegative functions such that
∑_{k}p_{k} = 1. We
obtain sufficient conditions for the existence of global positive
solutions to semilinear equations of this form. Our results extend
previous work by...
Allman, Elizabeth S.; Ané, Cécile; Rhodes, John A.
Inference of evolutionary trees and rates from biological
sequences is commonly performed using continuous-time Markov
models of character change. The Markov process evolves along an
unknown tree while observations arise only from the tips of the
tree. Rate heterogeneity is present in most real data sets and is
accounted for by the use of flexible mixture models where each
site is allowed its own rate. Very little has been rigorously
established concerning the identifiability of the models currently
in common use in data analysis, although nonidentifiability was
proven for a semiparametric model and an incorrect proof of
identifiability was published for a general parametric model
(GTR + Γ + I). Here...
Iksanov, Alex; Möhle, Martin
Let S_{0} := 0 and
S_{k} := ξ_{1} + ··· + ξ_{k}
for k ∈ N := {1, 2, ...}, where
{ξ_{k} : k ∈ N} are
independent copies of a random variable ξ with values in
N and distribution
p_{k} := P{ξ = k},
k ∈ N. We interpret the random walk
{S_{k} : k = 0, 1, 2, ...} as a
particle jumping to the right through integer positions. Fix
n ∈ N and modify the process by requiring
that the particle is bumped back to its current state each time a
jump would bring the particle to a state larger than or equal to
n. This constraint defines an increasing...
Hansen, Jennie C.; Jaworski, Jerzy
In this paper we investigate the `local' properties of a random
mapping model,
T_{n}^{D̂}, which maps
the set {1, 2, ..., n} into itself. The random mapping
T_{n}^{D̂}, which was
introduced in a companion paper (Hansen and Jaworski (2008)), is
constructed using a collection of exchangeable random variables
D̂_{1}, ..., D̂_{n}
which satisfy
∑_{i=1}^{n}D̂_{i} = n.
In the random digraph,
G_{n}^{D̂}, which
represents the mapping
T_{n}^{D̂}, the
in-degree sequence for the vertices is given by the variables
D̂_{1}, D̂_{2}, ..., D̂_{n},
and, in some sense,
G_{n}^{D̂} can be
viewed as an analogue of the general independent degree models
from random graph theory. By local properties we mean the
distributions of random mapping characteristics related to a given
vertex v of
G_{n}^{D̂} -...
Wong, Shek-Keung Tony
This paper revisits a general optimal stopping problem that often
appears as a special case in some finance applications. The
problem is essentially of the same form as the investment-timing
problem of McDonald and Siegel (1986) in which the underlying
processes are two correlated geometric Brownian motions (GBMs)
with drifts less than the discount rate. By contrast, we attempt
to analyze the underlying optimal stopping problem to its full
generality without imposing any restriction on the drifts of the
GBMs. By extending the first passage time approach of Xia and Zhou
(2007) to the current context, we manage to obtain a complete and
explicit characterization of the solution to the...
Alòs, Elisa; Ewald, Christian-Oliver
We prove that the Heston volatility is Malliavin differentiable
under the classical Novikov condition and give an explicit
expression for the derivative. This result guarantees the
applicability of Malliavin calculus in the framework of the Heston
stochastic volatility model. Furthermore, we derive conditions on
the parameters which assure the existence of the second Malliavin
derivative of the Heston volatility. This allows us to apply
recent results of Alòs (2006) in order to derive
approximate option pricing formulae in the context of the Heston
model. Numerical results are given.