arXiv subjects

Antoine Ayache

Publications and source records attributed to Antoine Ayache.

25 records · Page 2Linked to original sources

arXiv2013

Linear fractional stable motion: a wavelet estimator of the $\al$ parameter

Linear fractional stable motion, denoted by $\{X_{H,\al}(t)\}_{t\in \R}$, is one of the most classical stable processes; it depends on two parameters $H\in (0,1)$ and $\al\in (0,2)$. The parameter $H$ characterizes the self-similarity property of $\{X_{H,\al}(t)\}_{t\in \R}$ while the parameter $\al$ governs the tail heaviness of its finite dimensional distributions; throughout our article we assume that the latter distributions are symmetric, that $H>1/\al$ and that $H$ is known. We show that, on the interval $[0,1]$, the asymptotic behaviour of the maximum, at a given scale $j$, of absolute values of the wavelet coefficients of $\{X_{H,\al}(t)\}_{t\in \R}$, is of the same order as $2^{-j(H-1/\al)}$; then we derive from this result a strongly consistent (i.e. almost surely convergent) statistical estimator for the parameter $\al$.

math.ST↗

arXiv2012

Hölder regularity and series representation of a class of stochastic volatility models

Let $Φ:\R\rightarrow\R$ be an arbitrary continuously differentiable deterministic function such that $|Φ|+|Φ'|$ is bounded by a polynomial. In this article we consider the class of stochastic volatility models in which ${Z(t)}_{t\in [0,1]}$, the logarithm of the price process, is of the form $Z(t)=\int_{0}^t Φ(X(s)) dW(s)$, where ${X(s)}_{s\in[0,1]}$ denotes an arbitrary centered Gaussian process whose trajectories are, with probability 1, Hölder continuous functions of an arbitrary order $α\in (1/2,1]$, and where ${W(s)}_{s\in[0,1]}$ is a standard Brownian motion independent on ${X(s)}_{s\in [0,1]}$. First we show that the critical Hölder regularity of a typical trajectory of ${Z(t)}_{t\in[0,1]}$ is equal to 1/2. Next we provide for such a trajectory an expression as a random series which converges at a geometric rate in any Hölder space of an arbitrary order $γ<1/2$; this expression is obtained through the expansion of the random function $s\mapsto Φ(X(s))$ on the Haar basis. Finally, thanks to it, we give an efficient iterative simulation method for ${Z(t)}_{t\in[0,1]}$.

math.PR↗

arXiv2012

Sharp estimates on the tail behavior of a multistable distribution

Multistable distributions, which have been introduced recently by Falconer, Lévy Véhel and their co-authors, are natural generalizations of symmetric "alpha" stable distributions; roughly speaking, they are obtained by replacing the constant parameter "alpha" by a (Lebesgue) mesurable function. It is known that the tail of a symmetric "alpha" stable distribution asymptotically behaves as a power function with exponent "-alpha"; in this article we extend the latter result to the setting of multistable distributions.

math.PR↗

arXiv2012

Continuous Gaussian multifractional processes with random pointwise Hölder regularity

Let X be an arbitrary centered Gaussian process whose trajectories are, with probability one, continuous nowhere differentiable functions. It follows from a classical result, derived from zero-one law, that, with probability one, the trajectories of X have the same global Hölder regularity over any compact interval, that is the uniform Hölder exponent does not depend on the choice of a trajectory. A similar phenomenon happens with their local Hölder regularity measured through the local Hölder exponent. Therefore, it seems natural to ask the following question: does such a phenomenon also occur with their pointwise Hölder regularity measured through the pointwise Hölder exponent? In this article, using the framework of multifractional processes, we construct a family of counterexamples showing that the answer to this question is not always positive.

math.PR↗

arXiv2009

A process very similar to multifractional Brownian motion

In Ayache and Taqqu (2005), the multifractional Brownian (mBm) motion is obtained by replacing the constant parameter $H$ of the fractional Brownian motion (fBm) by a smooth enough functional parameter $H(.)$ depending on the time $t$. Here, we consider the process $Z$ obtained by replacing in the wavelet expansion of the fBm the index $H$ by a function $H(.)$ depending on the dyadic point $k/2^j$. This process was introduced in Benassi et al (2000) to model fBm with piece-wise constant Hurst index and continuous paths. In this work, we investigate the case where the functional parameter satisfies an uniform Hölder condition of order $β>\sup_{t\in \rit} H(t)$ and ones shows that, in this case, the process $Z$ is very similar to the mBm in the following senses: i) the difference between $Z$ and a mBm satisfies an uniform Hölder condition of order $d>\sup_{t\in \R} H(t)$; ii) as a by product, one deduces that at each point $t\in \R$ the pointwise Hölder exponent of $Z$ is $H(t)$ and that $Z$ is tangent to a fBm with Hurst parameter $H(t)$.

stat.ME↗

arXiv2008

Joint continuity of the local times of fractional Brownian sheets

Let $B^H=\{B^H(t),t\in{\mathbb{R}_+^N}\}$ be an $(N,d)$-fractional Brownian sheet with index $H=(H_1,...,H_N)\in(0,1)^N$ defined by $B^H(t)=(B^H_1(t),...,B^H_d(t)) (t\in {\mathbb{R}}_+^N),$ where $B^H_1,...,B^H_d$ are independent copies of a real-valued fractional Brownian sheet $B_0^H$. We prove that if $d<\sum_{\ell=1}^NH_{\ell}^{-1}$, then the local times of $B^H$ are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields 124 (2002)). We also establish sharp local and global Hölder conditions for the local times of $B^H$. These results are applied to study analytic and geometric properties of the sample paths of $B^H$.

math.PR↗

arXiv2008

Linear Fractional Stable Sheets: wavelet expansion and sample path properties

In this paper we give a detailed description of the random wavelet series representation of real-valued linear fractional stable sheet introduced in Ayache, Roueff and Xiao (2007). By using this representation, in the case where the sample paths are continuous, an anisotropic uniform and quasi-optimal modulus of continuity of these paths is obtained as well as an upper bound for their behavior at infinity and around the coordinate axes. The Hausdorff dimensions of the range and graph of these stable random fields are then derived.

math.ST↗