Abstract

Noisy diffusions are ubiquitous in applications—such as in physics, biology, finance, and atmosphere/ocean science. Most of the stochastic models that are used in applications involve unknown parameters which can be, in principle, estimated from observations of the Ito process. In many cases the observations of a diffusion process are contaminated by high-frequency observation error. It is therefore important to develop accurate and efficient statistical inference procedures that take into account this contamination of the observed high-frequency data, to ensure well-behaved procedures even in the limit of the sampling period tending to zero. Our goal is to address this issue by developing statistical inference methodologies in the frequency domain, in particular for estimating the integrated volatility.

We shall show that intuition and understanding can be found in the frequency domain, using the Fourier transform of the increments of the observed process. A shrinkage estimator will be proposed based on the sampling properties of this Fourier transform. The asymptotic variance of this new estimator will be derived, and the estimator will be shown to be asymptotically efficient. The estimator also has an interesting interpretation as smoothing the empirical auto-covariance sequence of the increments of the observations. The method can easily be generalized to the case of correlated observation error, and simulation studies illustrate a number of interesting properties of the estimation procedure.

Sofia Olhede (UCL), joint work with Greg Pavliotis and Adam Sykulski (Imperial)