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

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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.