Perturbation-based inference for diffusion processes: Obtaining effective models from multiscale data
For researchers working on parameter estimation in stochastic models from noisy or multiscale data, this work identifies a fundamental instability in standard methods and provides a convergent alternative.
The paper addresses parameter inference for stochastic differential equations when only perturbed observations are available. It shows that standard estimators like MLE are not convergent in this setting, and proposes a novel convergent inference procedure, demonstrated on multiscale data.
We consider the inference problem for parameters in stochastic differential equation models from discrete time observations (e.g. experimental or simulation data). Specifically, we study the case where one does not have access to observations of the model itself, but only to a perturbed version which converges weakly to the solution of the model. Motivated by this perturbation argument, we study the convergence of estimation procedures from a numerical analysis point of view. More precisely, we introduce appropriate consistency, stability, and convergence concepts and study their connection. It turns out that standard statistical techniques, such as the maximum likelihood estimator, are not convergent methodologies in this setting, since they fail to be stable. Due to this shortcoming, we introduce and analyse a novel inference procedure for parameters in stochastic differential equation models which turns out to be convergent. As such, the method is particularly suited for the estimation of parameters in effective (i.e. coarse-grained) models from observations of the corresponding multiscale process. We illustrate these theoretical findings via several numerical examples.