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Section: New Results

Inference for Gaussian systems

Participants: F. Baudoin, A. Chronopoulou, S. Cohen, F. Gamboa, Y. Hu, M. Jolis, C. Lacaux, J-M. Loubes, A. Neuenkirch, D. Nualart, C. Ouyang.

(i) LAN property for fractional Brownian motion. 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 (see [66] for details on this property). In [11] , 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 (X n ) n , whose spectral density f θ is indexed by a parameter θ, can admit antipersitence, long memory or short memory and be noninvertible. To be more specific, our main assumption is:

f θ (x) x0 |x| -α(θ) L θ (x)

with L θ a slowly varying function and α(θ)(-,1). We prove the LAN property by studying an asymptotic expansion of the log likelihood and using some results on Toeplitz matrices (see [39] , [53] ). 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.

(ii) 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 (-valued) equation handled up to now as far as parameter estimation is concerned (see [64] ) is of the form:

X t θ =a+θ 0 t b(X u )du+B t ,

where θ 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 n -valued equation:

X t θ =a+ 0 t b(θ;X u )du+ 0 t σ(θ;X u )dB t ,(2)

where θ enters non linearly in the coefficient, where σ 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 2011:

  • A better understanding of the underlying rough path structure for equation (2 ), carried out in [4] , [5] . This step allows a proper definition of our equation of interest in a wide range of contexts.

  • Gaussian type bounds for equations driven by a fractional Brownian motion, obtained in [9] . This is an important preliminary step for likelihood estimates for stochastic processes.

  • Numerical aspects of a maximum likelihood type procedure for an equation of the form (2 ), expressed in terms of Malliavin calculus tools (see [10] ).

  • Convergence of a least square type estimator for an equation of the form (2 ) where the noise enters additively, handled in [14] . This is the first occurrence of a converging estimator for a general coefficient b(θ,·).