Prediction-based inference for integrated diffusions with high-frequency data

Abstract We consider parametric inference for an ergodic and stationary diffusion process, when the data are high-frequency observations of the integral of the diffusion process. Such data are obtained via certain measurement devices, or if positions are recorded and speed is modelled by a diffusion. In finance, realized volatility or variations thereof can be used to construct observations of the latent integrated volatility process. Specifically, we assume that the integrated process is observed at n equidistant, deterministic time points i ⁢ Δ n {i\Delta_{n}} for some Δ n > 0 {\Delta_{n}>0} and consider the high-frequency/infinite horizon asymptotic scenario, where n → ∞ {n\to\infty} , Δ n → 0 {\Delta_{n}\to 0} and n ⁢ Δ n → ∞ {n\Delta_{n}\to\infty} . Subject to mild standard regularity conditions on ( X t ) {(X_{t})} , we prove the asymptotic existence and uniqueness of a consistent estimator for useful and tractable classes of prediction-based estimating functions. Asymptotic normality of the estimator is obtained under the additional rate assumption n ⁢ Δ n 2 → 0 {n\Delta_{n}^{2}\to 0} . The proofs are based on the useful Euler–Itô expansions of transformations of diffusions and integrated diffusions, which we study in some detail.