State and parameter estimation of stochastic physical systems from uncertain and indirect measurements

The approximate innovation methods have shown to be highly effective for the state and parameter estimation of a variety of continuous-time stochastic dynamical systems given a set of noisy discrete measurements. This article focuses on various issues of these inference methods not previously treated or discussed in deep such as the computation of the Fisher information matrix, the estimation of confidence intervals, and the evaluation of the fitting-innovation process as Gaussian white noise. These statistical tools play an essential role in practice as measures of goodness of fit of the model to the data and of the quality of the estimated parameters, which makes them suitable for designing of models and for optimal experimental design.