Section: New Results

Inference for Gaussian systems

Participants: T. Cass, S. Cohen, M. Hairer, C. Litterer, F. Panloup, L. Quer, S. Tindel.

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 (see [69] ) 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 2012:

• An implementable numerical scheme for equations driven by irregular processes, which is one of the ingredients one needs in order to perform an accurate statistical estimation procedure (see [6] ).

• A better understanding of the law of the solution $X_t^{\theta }$ to equation (1 ), carried out in [25] . This step allows to obtain smoothness of density for our equation of interest in a wide range of contexts, which is an essential prerequisite for a good estimation procedure.

• Another important preliminary step for likelihood estimates for stochastic equations is a good knowledge of their invariant measure in the ergodic case. This is the object of our article [27] .

• Finally we have also progressed in our knowledge of noisy differential systems by extending the range of applications of rough paths methods [11] , [14] .