Inference and Ill-Posedness for Financial High Frequency Data

近年来，高频金融数据的可用性呈爆炸式增长。 这为每天高精度地估计波动率等数量提供了可能性。 本计画系利用非参数“潜半鞅”模型，探讨高频资料之波动率及相关量之估计。“微观结构的存在（统计上相当于测量误差）对这个问题至关重要，因为它会严重影响估计。 早期的研究工作发现，估计是强大的添加剂的错误。 该项目关注的情况下，呼吁这种鲁棒性的问题，特别是，当价格的形成有一个四舍五入的组成部分，当有许多（也许无限多）的潜在过程中的小跳跃。 在这两种情况下，如何定义波动性有几种可能的候选者，并且过程和错误之间的分离并不总是很好地定义的。 这个项目将探讨这种不适定性，并寻求定义的波动性的措施，是强大的过程结构中的小变化。 该项目还将探讨这些措施的应用风险管理，预测，投资组合管理，期权交易，和regulation.This项目是既与估计和应用程序的金融实践和监管。 所提出的解决方案将依赖于统计和随机过程的技术，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。 鉴于波动率的估计是投资者，监管机构和政策制定者感兴趣的，本研究的结果将具有重大的实际意义。 该奖项是作为2006财政年度数学科学优先领域数学社会和行为科学（MSBS）特别竞赛的一部分获得支持的。

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).