A New Approach Toward Optimal and Adaptive Nonparametric Methods for High-Frequency Data

复杂系统的高频监测在神经科学、湍流和环境科学等许多领域都具有日益重要的意义。然而，随着交易数据集和限价订单数据集的出现，这种趋势在金融市场尤为突出，这两个数据集构成了统计学中“大数据”的两个典型例子。鉴于刚才提到的技术进步，基于高频观测的连续时间过程的统计推断方法在过去几年中得到了迅速的发展。该课题当前技术状态的关键问题之一在于，大多数提出的方法严重依赖于需要校准的调谐参数。当然，这是其他经典统计问题中使用的大多数非参数方法的情况。然而，在其他框架中解决这些问题的广泛文献尚未完全转化为随机过程的基于高频的推理。另一个重要问题来自于采用随机动力系统指定的人工模型的普遍做法，众所周知，这些模型在超高频下缺乏足够的精度来描述资产价格的风格化特征。这反过来又推动了微结构噪声概念的引入，但有时又假设了不自然的假设。因此，确实需要自下而上的模型推导，以便更好地理解基础资产价格的形成。首席调查员将利用上述需求，并有望在以下三个具有理论和实践意义的主要方向上显著推进这一领域：(1)设计新的方法，以实施与方法的内在调整参数有关的“最佳”推理方法；(2)开发基于数据驱动的定点程序的自适应估计方法的新方法以及必要的理论基础；(3)在基本近似跳跃扩散过程和微观结构噪声的建模中更好地结合极限顺序手册数据，以期增强基于极限顺序手册信息的潜在过程参数的估计。

High frequency monitoring of complex systems, which operate in continuous time, is increasingly important in many fields such as neural science, turbulence, and environmental sciences. This tendency, however, has been particularly predominant in financial markets with the advent of transaction and Limit Order Book data sets, which constitute two prototypical examples of "big data" in statistics. In light of the just mentioned technological advances, statistical inference methods for continuous time processes based on high-frequency observations have seen a rapid evolution in the last few years. One of the crucial issues with the current state of art of the subject lies in the fact that most of the proposed methods critically depend on tuning parameters that need to be calibrated. This, of course, is the case with most of the nonparametric methods used in other classical statistical problems. However, the extensive literature for resolving these problems in other frameworks has not yet been fully translated into the context of high-frequency-based inference for stochastic processes. Another important issue comes from the common practice of adopting artificial models specified by stochastic dynamical systems, which are known to lack sufficient accuracy for describing the stylized features of asset prices at ultra high frequency. This, in turn, has motivated the introduction of the concept of microstructure noise, but sometimes, again, assuming unnatural assumptions. Hence, there is a real need for bottom-up derivations of models that allow a better understanding of the underlying asset price formation.The principal investigator will tap on the previously mentioned needs and is expected to significantly advance the area in the following three primary directions of theoretical and practical relevance: (1) Devise new methodologies towards the implementation of "optimal" inference methods in regard to the intrinsic tuning parameters of the methods; (2) Develop a new approach, together with the necessary theoretical foundations, for adaptive estimation methods based on data-driven fixed point procedures; (3) Better incorporation of Limit Order Book data in the modeling of both the underlying approximating jump-diffusion process and the microstructure noise with a view of enhancing the estimation of latent process parameters based on limit order book information.