We consider estimation of scalar functions which determine the dynamics of diffusion processes. It has been recently shown that nonparametric maximum likelihood 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 specified in the function space by means of its precision operator, which is defined as an appropriate differential operator. We establish that a Bayesian Gaussian conjugate analysis for the drift of one-dimensional non-linear diffusions is feasible given high-frequency data. This is achieved by expressing the log-likelihood as a quadratic function of the drift, with sufficient statistics given by the so-called 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 non-parametric inference.
Summary. Hypoelliptic diffusion processes can be used to model a variety of phenomena in applications ranging from molecular dynamics to audio signal analysis. We study parameter estimation for such processes in situations where we observe some components of the solution at discrete times. Since exact likelihoods for the transition densities are typically not known, approximations are used that are expected to work well in the limit of small inter-sample times ∆t and large total observation times N ∆t. Hypoellipticity together with partial observation leads to ill-conditioning requiring a judicious combination of approximate likelihoods for the various parameters to be estimated. We combine these in a deterministic scan Gibbs sampler alternating between missing data in the unobserved solution components, and parameters. Numerical experiments illustrate asymptotic consistency of the method when applied to simulated data. The paper concludes with application of the Gibbs sampler to molecular dynamics data.
We study a Bayesian approach to nonparametric estimation of the periodic drift function of a one-dimensional diffusion from continuoustime data. Rewriting the likelihood in terms of local time of the process, and specifying a Gaussian prior with precision operator of differential form, we show that the posterior is also Gaussian with precision operator also of differential form. The resulting expressions are explicit and lead to algorithms which are readily implementable. Using new functional limit theorems for the local time of diffusions on the circle, we bound the rate at which the posterior contracts around the true drift function.
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