Section: New Results

Inference for gaussian systems

Participants: C. Lacaux, S. Tindel

Inference for dynamical systems driven by Gaussian noises.

As mentioned at the Scientific Foundations Section, the problem of estimating the coefficients of a general differential equation driven by a Gaussian process is still largely unsolved. To be more specific, the most general ($ℝ$-valued) equation handled up to now as far as parameter estimation is concerned is of the form:

where $\theta$ is the unknown parameter, $b$ is a smooth enough coefficient and $B$ is a one-dimensional fractional Brownian motion. In contrast with this simple situation, our applications of interest (see the Application Domains Section) require the analysis of the following ${ℝ}^n$-valued equation:

where $\theta$ enters non linearly in the coefficient, where $\sigma$ is a non-trivial diffusion term and $B$ is a $d$-dimensional fractional Brownian motion. We have thus decided to tackle this important scientific challenge first.

To this aim, here are the steps we have focused on in 2013:

• A better understanding of the underlying rough path structure for equation (4 ), carried out in [6] . This step allows a proper definition of stochastic integrals with respect to fractional Brownian motion in a wide range of contexts.

• Extension of pathwise stochastic integration to processes indexed by the plane in [19] , which helps to the definition of noisy systems such as partial differential equations.

• Gaussian type bounds for equations driven by a fractional Brownian motion, obtained in [18] , [7] . This is an important preliminary step for likelihood estimates for stochastic processes. Also notice the interesting central limit theorems exhibited in [24] , in a context which is similar to our equation of interest.

• Numerical aspects of a maximum likelihood type procedure for an equation of the form (4 ), expressed in terms of Malliavin calculus tools (see [4] ).

LAN property for fractional Brownian motion

We have first focused on an important statistical property of fractional Brownian paths on their own. Indeed, the local asymptotic normality (LAN) property is a fundamental concept in asymptotic statistics, which gives the asymptotic normality of certain estimators such as the maximum likelihood estimator for instance. In [5] , we focus on the LAN property for the model where we observe a sample of $n$ observations ${𝐗}_{𝐧}=(X_1,...,X_n)$ of a Gaussian stationary sequence. The sequence ${\left(X_n\right)}_{n\in \mathbb{N}}$, whose spectral density $f_{\theta }$ is indexed by a parameter $\theta$, can admit antipersistence, long memory or short memory and be noninvertible. To be more specific, our main assumption is:

with $L_{\theta }$ a slowly varying function and $\alpha \left(\theta \right)\in (-\infty ,1)$. We prove the LAN property by studying an asymptotic expansion of the log likelihood and using some results on Toeplitz matrices. In particular, our assumptions are fulfilled by fractional Gaussian noises and autoregressive fractionally integrated moving average processes (ARFIMA($p,d,q$)). We also obtain the LAN property for fractional Brownian motion.

Self-similarity properties and stable or Gaussian random fields

In 2009, C. Lacaux and H. Biermé carried on the study of some sample paths properties for an important class of anisotropic random fields called operator scaling random fields, which had been previously introduced by H. Biermé, M. Meerschaert and P. Scheffler (2007). To be more specific, the classical self-similarity property is replaced by the following operator scaling property:

where $c^E:=exp\left(Eln\left(c\right)\right).$ In particular, the Hölder regularity properties of operator scaling Gaussian or stable harmonizable random fields have been expressed in terms of the matrix $E$. The method they used can be applied to study the modulus of continuity of many stable or Gaussian random fields. As example in 2011, with P. Scheffler, they have followed it to study multi-operator harmonizable stable random fields, which satisfy a local version of the operator scaling property and enjoy a regularity which may vary along the trajectories. In [20] , it has been developed in the more general framework of conditionally sub-Gaussian random series. This allows to also study for example some multistable random fields, which have been introduced in (Falconer & al, 2009); for such a field $X$, the marginal $X\left(x\right)$ is a stable random variable whose index of stability can depend on $x$. In this paper, some conditions have been proposed to establish the uniform convergence of the series (on an eventually random ball), an upper bound for the modulus of continuity of its limit, an uniform control of the partial series ones and an explicit rate of convergence. Focusing on LePage random series, upper bounds of the modulus of continuity of some harmonizable stable or multistable random fields are provided. In the conference paper [11] , [20] has then been applied to study the class of linear multifractional multistable motions. In particular, the upper bound obtained for the modulus of linear multifractional stable motion is the sharpest available.

We are also interested in self-similar processes indexed by manifolds in [8] . This study is motivated by the fact various spatial data are indexed by a manifold and not by the Euclidean space ${ℝ}^d$ in practical situations such as image analysis.