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: 0806145
• 负责人: Jose Blanchet
• 金额: $ 28.2万
• 依托单位: Harvard 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: 2008-07-15 至 2008-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0806145&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.

模型构建通常由支持性能分析和分析计算的功能来指导。这种类型的方便特征的示例包括线性、高斯或光尾特征。研究人员打算开发数学工具，使随机系统的分析，表现出非线性，非高斯和潜在的重尾类型的特征。他们的目标不仅是提供工具，可用于确定何时高斯近似是适当的，以及在哪些空间尺度上应该使用大偏差或重尾渐近，但他们还旨在开发技术，以改善高斯和尾部渐近通过校正近似和尖锐的大偏差结果。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，允许了解这种空间尺度如何过渡到一个大的偏差区域，其中可能包含重尾近似和更精确的信息。由于系统的定性行为可以根据各种输入特性（例如，轻尾与重尾）而发生显着变化，因此当可以安全地使用易于处理的近似时，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/或甚至是重尾特征（如高斯过程和高斯过程）。例如，保险索赔或复杂金融证券的投资组合将其单个风险因素计算为数千个。这些因素可能会导致非常大的损失（重尾），这些因素之间的依赖结构非常复杂（导致非高斯行为），这在整体风险状况中至关重要。因此，这种复杂模型的分析在计算和分析上都是具有挑战性的，因此有必要诉诸近似和有效的计算算法。研究人员建议使用和发展数学技术，以更好地了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量损失或极端行为的分析类型。本研究的成果将改善复杂随机系统在上述领域的性能分析。