1,940 research outputs found
Efficient Rare-Event Simulation for Multiple Jump Events in Regularly Varying Random Walks and Compound Poisson Processes
We propose a class of strongly efficient rare event simulation estimators for
random walks and compound Poisson processes with a regularly varying
increment/jump-size distribution in a general large deviations regime. Our
estimator is based on an importance sampling strategy that hinges on the
heavy-tailed sample path large deviations result recently established in Rhee,
Blanchet, and Zwart (2016). The new estimators are straightforward to implement
and can be used to systematically evaluate the probability of a wide range of
rare events with bounded relative error. They are "universal" in the sense that
a single importance sampling scheme applies to a very general class of rare
events that arise in heavy-tailed systems. In particular, our estimators can
deal with rare events that are caused by multiple big jumps (therefore, beyond
the usual principle of a single big jump) as well as multidimensional processes
such as the buffer content process of a queueing network. We illustrate the
versatility of our approach with several applications that arise in the context
of mathematical finance, actuarial science, and queueing theory
Steady-state simulation of reflected Brownian motion and related stochastic networks
This paper develops the first class of algorithms that enable unbiased
estimation of steady-state expectations for multidimensional reflected Brownian
motion. In order to explain our ideas, we first consider the case of compound
Poisson (possibly Markov modulated) input. In this case, we analyze the
complexity of our procedure as the dimension of the network increases and show
that, under certain assumptions, the algorithm has polynomial-expected
termination time. Our methodology includes procedures that are of interest
beyond steady-state simulation and reflected processes. For instance, we use
wavelets to construct a piecewise linear function that can be guaranteed to be
within distance (deterministic) in the uniform norm to Brownian
motion in any compact time interval.Comment: Published at http://dx.doi.org/10.1214/14-AAP1072 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks
We consider a standard splitting algorithm for the rare-event simulation of
overflow probabilities in any subset of stations in a Jackson network at level
n, starting at a fixed initial position. It was shown in DeanDup09 that a
subsolution to the Isaacs equation guarantees that a subexponential number of
function evaluations (in n) suffice to estimate such overflow probabilities
within a given relative accuracy. Our analysis here shows that in fact
O(n^{2{\beta}+1}) function evaluations suffice to achieve a given relative
precision, where {\beta} is the number of bottleneck stations in the network.
This is the first rigorous analysis that allows to favorably compare splitting
against directly computing the overflow probability of interest, which can be
evaluated by solving a linear system of equations with O(n^{d}) variables.Comment: 23 page
Efficient rare-event simulation for multiple jump events in regularly varying random walks and compound Poisson processes
We propose a class of strongly efficient rare-event simulation estimators for random walks and compound Poisson processes with a regularly varying increment/jump-size distribution in a general large deviations regime. Our estimator is based on an importance sampling strategy that hinges on a recently established heavy-tailed sample-path large deviations result. The new estimators are straightforward to implement and can be used to systematically evaluate the probability of a wide range of rare events with bounded relative error. They are “universal” in the sense that a single importance sampling scheme applies to a very general class of rare events that arise in heavy-tailed systems. In particular, our estimators can deal with rare events that are caused by multiple big jumps (therefore, beyond the usual principle of a single big jump) as well as multidimensional processes such as the buffer content process of a queueing network. We illustrate the versatility of our approach with several applications that arise in the context of mathematical finance, actuarial science, and queueing theory
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