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.
