Inference and Ill-Posedness for Financial High Frequency Data

近年来，高频金融数据的可用性出现了爆炸式增长。这开启了以高精度估计每日波动率等数量的可能性。该项目关注的是使用非参数“潜在半鞅模型”对高频数据的波动性和相关数量的估计。微观结构（统计上等同于测量误差）的存在对这个问题至关重要，因为它实质上影响了估计器。研究者早期的工作已经发现估计器对加性误差具有鲁棒性。该项目关注的是对这种稳健性提出质疑的情况；特别是，当价格的形成有一个四舍五入的成分时，当潜在过程中有许多（也许是无限多）小的跳跃时。在这两种情况下，对于如何定义波动性有几种可能的候选，并且过程和错误之间的分离并不总是定义得很好。本项目将探索这种病态，并寻求定义对过程结构中的小变化具有健壮性的波动性度量。该项目还将探索这些措施在风险管理、预测、投资组合管理、期权交易和监管方面的应用。这个项目既涉及评估，也涉及金融实践和监管的应用。提出的解决方案将依赖于统计学和随机过程的技术，包括嵌入方法，预计这项研究将导致新的数学和计量经济学理论的发展。鉴于投资者、监管机构和政策制定者对波动率的估计感兴趣，本研究的结果将具有重大的实际意义。该奖项作为2006财政年度数学科学优先领域数学社会和行为科学（MSBS）特别竞赛的一部分得到支持。

Recent years have seen an explosion in the availability of high-frequency financial data. This has opened the possibility of estimating quantities like volatility on a daily basis with high precision. This project is concerned with the estimation of volatility and related quantities for high-frequency data, using a nonparametric "latent semi-martingale model." The existence of microstructure (statistically equivalent to measurement error) is crucial to this problem, as it substantially affects estimators. Earlier work by the investigator has found estimators that are robust to additive errors. The project is concerned with situations that call this robustness into question; in particular, when the formation of prices have a component of rounding, and when there are many (perhaps infinitely many) small jumps in the latent process. In both cases, there are several possible candidates for how to define volatility, and the separation between process and error is not always well defined. This project will explore this ill-posedness and seek to define measures of volatility that are robust to small alterations in the process structure. The project also will explore the application of such measures to risk management, forecasting, portfolio management, options trading, and regulation.This project is concerned with both estimation and with applications to financial practice and regulation. The solution proposed will rely on techniques from statistics and stochastic processes, including embedding methods, and it is expected that this research will lead to the development of new mathematical and econometric theory. Given that estimates of volatility are of interest to investors, regulators, and policymakers, the results of this research will have substantial practical implications. This award was supported as part of the fiscal year 2006 Mathematical Science priority area special competition on Mathematical Social and Behavioral Sciences (MSBS).