Inference and Ill-Posedness for Financial High Frequency Data

近年来，高频金融数据的可获得性呈爆炸式增长。这为每天高精度地估计波动率等数量提供了可能性。这个项目是关于高频数据的波动率和相关数量的估计，使用一个非参数的“潜在半鞅模型”。微观结构(在统计上等同于测量误差)的存在对这个问题至关重要，因为它对估计者有很大的影响。研究人员的早期工作已经找到了对加性误差具有稳健性的估计器。该项目涉及的情况使这种稳健性受到质疑；特别是当价格的形成具有舍入成分时，以及在潜在过程中有许多(可能是无穷多的)小跳跃。在这两种情况下，如何定义波动性都有几种可能的候选者，过程和错误之间的分离并不总是被很好地定义。该项目将探索这一不适定性，并寻求定义对流程结构中的微小变化具有健壮性的波动性衡量标准。该项目还将探索将这些措施应用于风险管理、预测、投资组合管理、期权交易和监管。该项目涉及评估以及在金融实践和监管中的应用。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。鉴于投资者、监管者和政策制定者对波动率的估计很感兴趣，这项研究的结果将具有重大的实际意义。该奖项是2006财政年度数学科学优先领域数学社会科学和行为科学特别竞赛的一部分。

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).