Parameter estimation of stochastic chaotic systems

A standard way of formulating stochastic differential equation systems is to additively extend the drift of a deterministic system with a random diffusion part, but an increasing trend in applications, such as meteorology, is to perturb deterministic models in multiplicative and highly nonlinear ways, that escape the standard framework. This work aims to present a Bayesian method that enables estimating the parameters of such systems. The approach is well suited for situations were the observation times are irregular with large gaps between, so that the use of usual prediction-based filtering methods is excluded. The key idea is to construct a likelihood that is based on feature vectors that characterize the variability of the system. We illustrate the capability of the method in different scenarios that are both chaotic and stochastic using the classical Lorenz system as the demonstration example.