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

Noncentral convergence of multiple integrals

Fix $\nu>0$, denote by $G(\nu/2)$ a Gamma random variable with parameter $\nu/2$ and let $n\geq2$ be a fixed even integer. Consider a sequence $\{F_k\}_{k\geq1}$ of square integrable random variables belonging to the $n$th Wiener chaos of a given Gaussian process and with variance converging to $2\nu$. As $k\to\infty$, we prove that $F_k$ converges in distribution to $2G(\nu/2)-\nu$ if and only if $E(F_k^4)-12E(F_k^3)\to12\nu^2-48\nu$.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

Universal Gaussian fluctuations of non-Hermitian matrix ensembles: from weak convergence to almost sure CLTs

In the paper [25], written in collaboration with Gesine Reinert, we proved a universality principle for the Gaussian Wiener chaos. In the present work, we aim at providing an original example of application of this principle in the framework of random matrix theory. More specifically, by combining the result in [25] with some combinatorial estimates, we are able to prove multi-dimensional central limit theorems for the spectral moments (of arbitrary degrees) associated with random matrices with real-valued i.i.d. entries, satisfying some appropriate moment conditions. Our approach has the advantage of yielding, without extra effort, bounds over classes of smooth (i.e., thrice differentiable) functions, and it allows to deal directly with discrete distributions. As a further application of our estimates, we provide a new "almost sure central limit theorem", involving logarithmic means of functions of vectors of traces.Comment: 40 pages. This is an expanded version of a paper formerly called "Universal Gaussian fluctuations of non-Hermitian matrix ensembles", by the same authors. Sections 1.4 and 5 (about almost sure central limit theorems) are ne

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

Correcting Newton--C\^{o}tes integrals by L\'{e}vy areas

In this note we introduce the notion of Newton--C\^{o}tes functionals corrected by L\'{e}vy areas, which enables us to consider integrals of the type $\int f(y) \mathrm{d}x,$ where $f$ is a ${\mathscr{C}}^{2m}$ function and $x,y$ are real H\"{o}lderian functions with index $\alpha>1/(2m+1)$ for all $m\in {\mathbb{N}}^*.$ We show that this concept extends the Newton--C\^{o}tes functional introduced in Gradinaru et al., to a larger class of integrands. Then we give a theorem of existence and uniqueness for differential equations driven by $x$, interpreted using the symmetric Russo--Vallois integral.Comment: Published at http://dx.doi.org/10.3150/07-BEJ6015 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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 $B$ with Hurst index $H$. In the quadratic (resp. cubic) case, when $H<1/4$ (resp. $H<1/6$), we show by means of Malliavin calculus that the convergence holds in $L^2$ toward an explicit limit which only depends on $B$. 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

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

Stochastic derivatives for fractional diffusions

In this paper, we introduce some fundamental notions related to the so-called stochastic derivatives with respect to a given $\sigma$-field $\mathcal{Q}$. In our framework, we recall well-known results about Markov--Wiener diffusions. We then focus mainly on the case where $X$ is a fractional diffusion and where $\mathcal{Q}$ is the past, the future or the present of $X$. We treat some crucial examples and our main result is the existence of stochastic derivatives with respect to the present of $X$ when $X$ solves a stochastic differential equation driven by a fractional Brownian motion with Hurst index $H>1/2$. We give explicit formulas.Comment: Published in at http://dx.doi.org/10.1214/009117906000001169 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org