1,283 research outputs found
A Dirichlet process characterization of a class of reflected diffusions
For a class of stochastic differential equations with reflection for which a
certain continuity condition holds with , it is shown
that any weak solution that is a strong Markov process can be decomposed into
the sum of a local martingale and a continuous, adapted process of zero
-variation. When , this implies that the reflected diffusion is a
Dirichlet process. Two examples are provided to motivate such a
characterization. The first example is a class of multidimensional reflected
diffusions in polyhedral conical domains that arise as approximations of
certain stochastic networks, and the second example is a family of
two-dimensional reflected diffusions in curved domains. In both cases, the
reflected diffusions are shown to be Dirichlet processes, but not
semimartingales.Comment: Published in at http://dx.doi.org/10.1214/09-AOP487 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Non-Parametric Estimation of Spot Covariance Matrix with High-Frequency Data
Estimating spot covariance is an important issue to study, especially with
the increasing availability of high-frequency financial data. We study the
estimation of spot covariance using a kernel method for high-frequency data. In
particular, we consider first the kernel weighted version of realized
covariance estimator for the price process governed by a continuous
multivariate semimartingale. Next, we extend it to the threshold kernel
estimator of the spot covariances when the underlying price process is a
discontinuous multivariate semimartingale with finite activity jumps. We derive
the asymptotic distribution of the estimators for both fixed and shrinking
bandwidth. The estimator in a setting with jumps has the same rate of
convergence as the estimator for diffusion processes without jumps. A
simulation study examines the finite sample properties of the estimators. In
addition, we study an application of the estimator in the context of covariance
forecasting. We discover that the forecasting model with our estimator
outperforms a benchmark model in the literature
Asymptotic approximations for stationary distributions of many-server queues with abandonment
A many-server queueing system is considered in which customers arrive
according to a renewal process and have service and patience times that are
drawn from two independent sequences of independent, identically distributed
random variables. Customers enter service in the order of arrival and are
assumed to abandon the queue if the waiting time in queue exceeds the patience
time. The state of the system with servers is represented by a
four-component process that consists of the forward recurrence time of the
arrival process, a pair of measure-valued processes, one that keeps track of
the waiting times of customers in queue and the other that keeps track of the
amounts of time customers present in the system have been in service and a
real-valued process that represents the total number of customers in the
system. Under general assumptions, it is shown that the state process is a
Feller process, admits a stationary distribution and is ergodic. It is also
shown that the associated sequence of scaled stationary distributions is tight,
and that any subsequence converges to an invariant state for the fluid limit.
In particular, this implies that when the associated fluid limit has a unique
invariant state, then the sequence of stationary distributions converges, as
, to the invariant state. In addition, a simple example is
given to illustrate that, both in the presence and absence of abandonments, the
and limits cannot always be
interchanged.Comment: Published in at http://dx.doi.org/10.1214/10-AAP738 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Fluid limits of many-server queues with reneging
This work considers a many-server queueing system in which impatient
customers with i.i.d., generally distributed service times and i.i.d.,
generally distributed patience times enter service in the order of arrival and
abandon the queue if the time before possible entry into service exceeds the
patience time. The dynamics of the system is represented in terms of a pair of
measure-valued processes, one that keeps track of the waiting times of the
customers in queue and the other that keeps track of the amounts of time each
customer being served has been in service. Under mild assumptions, essentially
only requiring that the service and reneging distributions have densities, as
both the arrival rate and the number of servers go to infinity, a law of large
numbers (or fluid) limit is established for this pair of processes. The limit
is shown to be the unique solution of a coupled pair of deterministic integral
equations that admits an explicit representation. In addition, a fluid limit
for the virtual waiting time process is also established. This paper extends
previous work by Kaspi and Ramanan, which analyzed the model in the absence of
reneging. A strong motivation for understanding performance in the presence of
reneging arises from models of call centers.Comment: Published in at http://dx.doi.org/10.1214/10-AAP683 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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