Nonparametric estimation of diffusions: a differential equations approach

We consider estimation of scalar functions that determine the dynamics of diffusion processes. It has been recently shown that nonparametric maximum likelihood estimation is ill-posed in this context. We adopt a probabilistic approach to regularize the problem by the adoption of a prior distribution for the unknown functional. A Gaussian prior measure is chosen in the function space by specifying its precision operator as an appropriate differential operator. We establish that a Bayesian-Gaussian conjugate analysis for the drift of one-dimensional nonlinear diffusions is feasible using high-frequency data, by expressing the loglikelihood as a quadratic function of the drift, with sufficient statistics given by the local time process and the end points of the observed path. Computationally efficient posterior inference is carried out using a finite element method. We embed this technology in partially observed situations and adopt a data augmentation approach whereby we iteratively generate missing data paths and draws from the unknown functional. Our methodology is applied to estimate the drift of models used in molecular dynamics and financial econometrics using high- and low-frequency observations. We discuss extensions to other partially observed schemes and connections to other types of nonparametric inference.