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

模型构建通常由支持性能分析和分析计算的特性来指导。这类方便特征的例子包括线性、高斯或轻尾特征。研究人员打算开发数学工具，以分析表现出非线性、非高斯和潜在重尾型特征的随机系统。他们的目标不仅是提供工具，可以用来确定何时高斯近似是合适的，在哪个空间尺度上应该使用大偏差或重尾渐近，而且他们还旨在开发技术，通过校正的近似和尖锐的大偏差结果来改进高斯和尾渐近。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，使人们能够理解这种空间尺度如何过渡到一个大偏差区域，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以根据不同的输入特性（例如，轻尾与重尾）发生巨大变化，因此确定可以安全地使用易于处理的近似的区域或尺度将具有很大的价值。通信网络、巨灾建模、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ARCH和GARCH型过程）。例如，保险索赔或复杂金融证券的投资组合以数千计计算其个人风险因素。这些因素可能导致极大的损失（重尾），这些因素之间的依赖关系结构非常复杂（导致非高斯行为），这对整体风险状况至关重要。因此，对这种复杂模型的分析在计算和分析上都具有挑战性，因此有必要采用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，以更好地理解基于高斯定律和线性化的标准近似何时适用；当必须考虑非线性特征时，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。本研究的结果将在上述领域改进复杂随机系统的性能分析。

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.