3,335 research outputs found

### Moderate deviations via cumulants

The purpose of the present paper is to establish moderate deviation
principles for a rather general class of random variables fulfilling certain
bounds of the cumulants. We apply a celebrated lemma of the theory of large
deviations probabilities due to Rudzkis, Saulis and Statulevicius. The examples
of random objects we treat include dependency graphs, subgraph-counting
statistics in Erd\H{o}s-R\'enyi random graphs and $U$-statistics. Moreover, we
prove moderate deviation principles for certain statistics appearing in random
matrix theory, namely characteristic polynomials of random unitary matrices as
well as the number of particles in a growing box of random determinantal point
processes like the number of eigenvalues in the GUE or the number of points in
Airy, Bessel, and $\sin$ random point fields.Comment: 24 page

### Moderate deviations for particle filtering

Consider the state space model (X_t,Y_t), where (X_t) is a Markov chain, and
(Y_t) are the observations. In order to solve the so-called filtering problem,
one has to compute L(X_t|Y_1,...,Y_t), the law of X_t given the observations
(Y_1,...,Y_t). The particle filtering method gives an approximation of the law
L(X_t|Y_1,...,Y_t) by an empirical measure \frac{1}{n}\sum_1^n\delta_{x_{i,t}}.
In this paper we establish the moderate deviation principle for the empirical
mean \frac{1}{n}\sum_1^n\psi(x_{i,t}) (centered and properly rescaled) when the
number of particles grows to infinity, enhancing the central limit theorem.
Several extensions and examples are also studied.Comment: Published at http://dx.doi.org/10.1214/105051604000000657 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org

### Moderate deviations for recursive stochastic algorithms

We prove a moderate deviation principle for the continuous time interpolation
of discrete time recursive stochastic processes. The methods of proof are
somewhat different from the corresponding large deviation result, and in
particular the proof of the upper bound is more complicated. The results can be
applied to the design of accelerated Monte Carlo algorithms for certain
problems, where schemes based on moderate deviations are easier to construct
and in certain situations provide performance comparable to those based on
large deviations.Comment: Submitted to Stochastic System

### Option Pricing in the Moderate Deviations Regime

We consider call option prices in diffusion models close to expiry, in an
asymptotic regime ("moderately out of the money") that interpolates between the
well-studied cases of at-the-money options and out-of-the-money fixed-strike
options. First and higher order small-time moderate deviation estimates of call
prices and implied volatility are obtained. The expansions involve only simple
expressions of the model parameters, and we show in detail how to calculate
them for generic local and stochastic volatility models. Some numerical
examples for the Heston model illustrate the accuracy of our results

### Moderate deviations and extinction of an epidemic

Consider an epidemic model with a constant flux of susceptibles, in a
situation where the corresponding deterministic epidemic model has a unique
stable endemic equilibrium. For the associated stochastic model, whose law of
large numbers limit is the deterministic model, the disease free equilibrium is
an absorbing state, which is reached soon or later by the process. However, for
a large population size, i.e. when the stochastic model is close to its
deterministic limit, the time needed for the stochastic perturbations to stop
the epidemic may be enormous. In this paper, we discuss how the Central Limit
Theorem, Moderate and Large Deviations allow us to give estimates of the
extinction time of the epidemic, depending upon the size of the population

### Moderate deviations for diffusions with Brownian potentials

We present precise moderate deviation probabilities, in both quenched and
annealed settings, for a recurrent diffusion process with a Brownian potential.
Our method relies on fine tools in stochastic calculus, including Kotani's
lemma and Lamperti's representation for exponential functionals. In particular,
our result for quenched moderate deviations is in agreement with a recent
theorem of Comets and Popov [Probab. Theory Related Fields 126 (2003) 571-609]
who studied the corresponding problem for Sinai's random walk in random
environment.Comment: Published at http://dx.doi.org/10.1214/009117904000000829 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org

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