AMC-SS: Collaborative Research: Stochastic Processes and Time Series Models: Algorithms, Asymptotics, and Phase Transitions

AMC-SS：协作研究：随机过程和时间序列模型：算法、渐近和相变

• 批准号: 0902075
• 负责人: Jose Blanchet
• 金额: $ 28.2万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0902075&HistoricalAwards=false
• 关键词: AMC; SS; Collaborative; Research; Stochastic

Model building is often guided by features that enable performance analysis and analytic computations. Examples of such types of convenient features include linearity, Gaussian or light-tailed features. The investigators intend to develop mathematical tools that enable the analysis of stochastic systems that exhibit non-linear, non-Gaussian and potentially heavy-tailed type characteristics. Their goal is not only to provide tools that can be used to identify when Gaussian approximations are appropriate and at which spatial scales large deviations or heavy-tailed asymptotics should be used, but they also aim to develop techniques to improve upon Gaussian and tail asymptotics by means of corrected approximations and sharp large deviation results. These techniques will help researchers identify the spatial and temporal scales under which Gaussian approximations are valid. In addition, the investigators aim to provide tools that allow to understand how such spatial scales transition into a large deviations region which may incorporate heavy-tailed approximations and more refined information. Since the qualitative behavior of a system can change dramatically depending upon various input characteristics (e.g. light vs. heavy-tailed), identifying regions or scales when tractable approximations can be safely used would be of great value. Recent developments in areas such as Communication Networks, Catastrophe Modeling, Insurance and Finance demand more complex time-series models that are either non-linear or exhibit non-Gaussian and/or even heavy-tailed features (such as ARCH and GARCH type processes). For example, portfolios of insurance claims or complex financial securities count their individual risk factors in the order of thousands. The factors can give rise to extremely large losses (heavy-tails) and the dependence structures among such factors, which is crucial in the overall risk profile, is very complex (giving rise to non-Gaussian behavior). As a consequence, the analysis of such complex models is challenging both computationally and analytically and therefore it is necessary to resort to approximations and efficient computational algorithms. The investigators propose the use and development of mathematical techniques to better understand when standard approximations, based on Gaussian laws and linearization, are applicable; when non-linear features must be taken into account and how does one transition from Gaussian-type approximations to a type of analysis that involves large losses or extreme behavior. The outcome of this research will improve the performance analysis of complex stochastic systems in the areas indicated above.

模型构建通常以支持性能分析和分析计算的功能为指导。此类方便特征的示例包括线性、高斯或轻尾特征。研究人员打算开发数学工具，能够分析表现出非线性、非高斯和潜在重尾类型特征的随机系统。他们的目标不仅是提供可用于识别高斯近似何时合适以及应使用大偏差或重尾渐近的空间尺度的工具，而且还旨在开发通过以下方式改进高斯和尾渐近的技术：修正近似值和尖锐的大偏差结果的方法。这些技术将帮助研究人员确定高斯近似有效的空间和时间尺度。此外，研究人员的目标是提供工具，使人们能够了解这种空间尺度如何转变为大偏差区域，该区域可能包含重尾近似和更精细的信息。由于系统的定性行为可能会根据各种输入特征（例如轻尾与重尾）而发生巨大变化，因此在可以安全使用易于处理的近似值时识别区域或尺度将具有很大的价值。 通信网络、灾难建模、保险和金融等领域的最新发展需要更复杂的时间序列模型，这些模型要么是非线性的，要么表现出非高斯和/或什至重尾特征（例如 ARCH 和 GARCH 类型过程） 。例如，保险索赔或复杂金融证券的投资组合将其各自的风险因素计算为数千个。这些因素可能会导致极大的损失（重尾），并且这些因素之间的依赖结构（这对于整体风险状况至关重要）非常复杂（导致非高斯行为）。因此，这种复杂模型的分析在计算和分析上都具有挑战性，因此有必要诉诸近似和高效的计算算法。研究人员建议使用和开发数学技术，以更好地理解基于高斯定律和线性化的标准近似何时适用；何时必须考虑非线性特征，以及如何从高斯型近似过渡到涉及大损失或极端行为的分析类型。这项研究的成果将改善上述领域的复杂随机系统的性能分析。