Filtered data and eigenfunction estimators for statistical inference of multiscale and interacting diffusion processes

We study the problem of learning unknown parameters of stochastic dynamical models from data. Often, these models are high dimensional and contain several scales and complex structures. One is then interested in obtaining a reduced, coarse-grained description of the dynamics that is valid at macroscopic scales. In this thesis, we consider two stochastic models: multiscale Langevin diffusions and noisy interacting particle systems. In both cases, a simplified description of the model is available through the theory of homogenization and the mean field limit, respectively. Inferring parameters in coarse-grained models using data from the full dynamics is a challenging problem since data are compatible with the surrogate model only at the macroscopic scale.In the first part of the thesis we consider the framework of overdamped two-scale Langevin equation and aim to fit effective dynamics from continuous observations of the multiscale model. In this setting, estimating parameters of the homogenized equation requires preprocessing of the data, often in the form of subsampling, because traditional maximum likelihood estimators fail. Indeed, they are asymptotically biased in the limit of infinite data and when the multiscale parameter vanishes. We avoid subsampling and work instead with filtered data, found by application of an appropriate kernel of the exponential family and a moving average. We then derive modified maximum likelihood estimators based on the filtered process, and show that they are asymptotically unbiased with respect to the homogenized equation. A series of numerical experiments demonstrate that our new approach allows to successfully infer effective diffusions, and that it is an improvement of traditional methods such as subsampling. In particular, our methodology is more robust, requires less knowledge of the full model, and is easy to implement. We conclude the first part presenting novel theoretical results about multiscale Langevin dynamics and proposing possible developments of the filtering approach.In the second part of the thesis we consider both multiscale diffusions and interacting particle systems, and we employ a different technique which is suitable for parameter estimation when a sequence of discrete observations is given. In particular, our estimators are defined as the zeros of appropriate martingale estimating functions constructed with the eigenvalues and the eigenfunctions of the generator of the effective dynamics. We first prove homogenization results for the generator of the multiscale Langevin equation and then apply our novel eigenfunction estimators to the two problems under investigation. Moreover, in the case of multiscale diffusions, we combine this strategy with the filtering methodology previously introduced. We prove that our estimators are asymptotically unbiased and present a series of numerical experiments which corroborates our theoretical findings, illustrates the advantages of our approach, and shows that our methodology can be employed to accurately fit simple models from complex phenomena.