224 research outputs found
A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one
In this paper, we will focus - in dimension one - on the SDEs of the type
dX_t=s(X_t)dB_t+b(X_t)dt where B is a fractional Brownian motion. Our principal
motivation is to describe one of the simplest theory - from our point of view -
allowing to study this SDE, and this for any Hurst index H between 0 and 1. We
will consider several definitions of solution and we will study, for each one
of them, in which condition one has existence and uniqueness. Finally, we will
examine the convergence or not of the canonical scheme associated to our SDE,
when the integral with respect to fBm is defined using the Russo-Vallois
symmetric integral
Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion
The present article is devoted to a fine study of the convergence of
renormalized weighted quadratic and cubic variations of a fractional Brownian
motion with Hurst index . In the quadratic (resp. cubic) case, when
(resp. ), we show by means of Malliavin calculus that the
convergence holds in toward an explicit limit which only depends on .
This result is somewhat surprising when compared with the celebrated Breuer and
Major theorem.Comment: Published in at http://dx.doi.org/10.1214/07-AOP385 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Noncentral convergence of multiple integrals
Fix , denote by a Gamma random variable with parameter
and let be a fixed even integer. Consider a sequence
of square integrable random variables belonging to the th
Wiener chaos of a given Gaussian process and with variance converging to
. As , we prove that converges in distribution to
if and only if .Comment: Published in at http://dx.doi.org/10.1214/08-AOP435 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Central limit theorems for multiple Skorohod integrals
In this paper, we prove a central limit theorem for a sequence of iterated
Shorohod integrals using the techniques of Malliavin calculus. The convergence
is stable, and the limit is a conditionally Gaussian random variable. Some
applications to sequences of multiple stochastic integrals, and renormalized
weighted Hermite variations of the fractional Brownian motion are discussed.Comment: 32 pages; major changes in Sections 4 and
Stein's method meets Malliavin calculus: a short survey with new estimates
We provide an overview of some recent techniques involving the Malliavin
calculus of variations and the so-called ``Stein's method'' for the Gaussian
approximations of probability distributions. Special attention is devoted to
establishing explicit connections with the classic method of moments: in
particular, we use interpolation techniques in order to deduce some new
estimates for the moments of random variables belonging to a fixed Wiener
chaos. As an illustration, a class of central limit theorems associated with
the quadratic variation of a fractional Brownian motion is studied in detail.Comment: 31 pages. To appear in the book "Recent advances in stochastic
dynamics and stochastic analysis", published by World Scientifi
On the Gaussian approximation of vector-valued multiple integrals
By combining the findings of two recent, seminal papers by Nualart, Peccati
and Tudor, we get that the convergence in law of any sequence of vector-valued
multiple integrals towards a centered Gaussian random vector , with
given covariance matrix , is reduced to just the convergence of: the
fourth cumulant of each component of to zero; the covariance
matrix of to . The aim of this paper is to understand more deeply this
somewhat surprising phenomenom. To reach this goal, we offer two results of
different nature. The first one is an explicit bound for in terms of
the fourth cumulants of the components of , when is a -valued
random vector whose components are multiple integrals of possibly different
orders, is the Gaussian counterpart of (that is, a Gaussian centered
vector sharing the same covariance with ) and stands for the Wasserstein
distance. The second one is a new expression for the cumulants of as above,
from which it is easy to derive yet another proof of the previously quoted
result by Nualart, Peccati and Tudor.Comment: 18 page
Exchangeable pairs on Wiener chaos
In [14], Nourdin and Peccati combined the Malliavin calculus and Stein's
method of normal approximation to associate a rate of convergence to the
celebrated fourth moment theorem [19] of Nualart and Peccati. Their analysis,
known as the Malliavin-Stein method nowadays, has found many applications
towards stochastic geometry, statistical physics and zeros of random
polynomials, to name a few. In this article, we further explore the relation
between these two fields of mathematics. In particular, we construct
exchangeable pairs of Brownian motions and we discover a natural link between
Malliavin operators and these exchangeable pairs. By combining our findings
with E. Meckes' infinitesimal version of exchangeable pairs, we can give
another proof of the quantitative fourth moment theorem. Finally, we extend our
result to the multidimensional case.Comment: 19 pages, submitte
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