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 equidistant, deterministic time points and consider the high-frequency/infinite horizon asymptotic scenario, where the number of observations, the sampling frequency and the time of the last observation all go to infinity. Subject to mild standard regularity conditions on the diffusion model, 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 an additional assumption on the rates. The proofs are based on the useful Euler-Ito expansions of transformations of diffusions and integrated diffusions, which we study in some detail.

翻译：我们考虑遍历平稳扩散过程的参数推断问题，其数据为扩散过程积分的高频观测值。此类数据可通过特定测量设备获得，或在记录位置且以扩散过程建模速度的情形下产生。在金融领域，已实现波动率或其变体可用于构建潜在积分波动率过程的观测值。具体而言，我们假设积分过程在等距确定性时间点被观测，并考虑高频/无限时域渐近场景——观测数量、采样频率及末次观测时间均趋于无穷。在扩散模型满足温和标准正则性条件下，我们证明了基于预测估计函数的有用且易处理类中一致估计量的渐近存在性与唯一性。在附加速率假设下获得了估计量的渐近正态性。证明基于扩散过程及积分扩散变换的实用欧拉-伊藤展开式，我们对此进行了详细研究。