411 research outputs found
Exact Simulation of Non-stationary Reflected Brownian Motion
This paper develops the first method for the exact simulation of reflected
Brownian motion (RBM) with non-stationary drift and infinitesimal variance. The
running time of generating exact samples of non-stationary RBM at any time
is uniformly bounded by where is the
average drift of the process. The method can be used as a guide for planning
simulations of complex queueing systems with non-stationary arrival rates
and/or service time
Shape-constrained Estimation of Value Functions
We present a fully nonparametric method to estimate the value function, via
simulation, in the context of expected infinite-horizon discounted rewards for
Markov chains. Estimating such value functions plays an important role in
approximate dynamic programming and applied probability in general. We
incorporate "soft information" into the estimation algorithm, such as knowledge
of convexity, monotonicity, or Lipchitz constants. In the presence of such
information, a nonparametric estimator for the value function can be computed
that is provably consistent as the simulated time horizon tends to infinity. As
an application, we implement our method on price tolling agreement contracts in
energy markets
Measuring the Initial Transient: Reflected Brownian Motion
We analyze the convergence to equilibrium of one-dimensional reflected
Brownian motion (RBM) and compute a number of related initial transient
formulae. These formulae are of interest as approximations to the initial
transient for queueing systems in heavy traffic, and help us to identify
settings in which initialization bias is significant. We conclude with a
discussion of mean square error for RBM. Our analysis supports the view that
initial transient effects for RBM and related models are typically of modest
size relative to the intrinsic stochastic variability, unless one chooses an
especially poor initialization.Comment: 14 pages, 3 figure
Central Limit Theorems and Large Deviations for Additive Functionals of Reflecting Diffusion Processes
This paper develops central limit theorems (CLT's) and large deviations
results for additive functionals associated with reflecting diffusions in which
the functional may include a term associated with the cumulative amount of
boundary reflection that has occurred. Extending the known central limit and
large deviations theory for Markov processes to include additive functionals
that incorporate boundary reflection is important in many applications settings
in which reflecting diffusions arise, including queueing theory and economics.
In particular, the paper establishes the partial differential equations that
must be solved in order to explicitly compute the mean and variance for the
CLT, as well as the associated rate function for the large deviations
principle
Tail asymptotics for the maximum of perturbed random walk
Consider a random walk that is ``perturbed'' by a
stationary sequence to produce the process
. This paper is concerned with computing the distribution
of the all-time maximum of perturbed
random walk with a negative drift. Such a maximum arises in several different
applications settings, including production systems, communications networks
and insurance risk. Our main results describe asymptotics for
as . The tail asymptotics depend greatly
on whether the 's are light-tailed or heavy-tailed. In the light-tailed
setting, the tail asymptotic is closely related to the Cram\'{e}r--Lundberg
asymptotic for standard random walk.Comment: Published at http://dx.doi.org/10.1214/105051606000000268 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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