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.
Stochastic models are used widely in engineering and social sciences to capture random phenomena. For instance, price returns are often modeled as random walks; traffic of information in networks are modeled using so-called queueing systems (or waiting line systems), etc. Gaussian distributions and processes are used to approximate the behavior of these systems. Gaussian models have convenient features that make them attractive (symmetry, parsimonious representations of risk in terms of variances etc.). The use of Gaussian distributions as approximations are justified in some cases but not in many others. The aim of this proposal is to develop methodology that describes when Gaussian approximations are applicable (this is made matematically rigorous by introducing scaling parameters) and what happens when these approximations are not applicable. That is, what are the right approximations to systems or how to correct the Gaussian approximations. Our results provide qualitative guidance in situations that are somewhat related to random walk dynamics. For instance, we consider random walks with power-law (i.e. heavy-tailed) decaying density of the increments. It is known that when there is finite variability Gaussian approximations are appropriate for ranges that increase like the square root of the amount of time that one observes the random walk. Gaussian descriptions have an error, which can be computed up to certain degree that relates to the power-law decay mentioned earlier. One of our papers extends the analysis of this error and we find that the correct description of the error blends nicely with so-called large deviations results. In practical terms: it is OK to use Gaussian approximations if you know corrections in the Gaussian regions and blend them naturally with rare-event asymptotics. This is useful because in principle one might think that for high fidelity approximations there might be something between the "regular zone" and the "large deviations zone" and we see that in the presence of power-law tail and random walk type models, this is not really the case. Another direction that we explore in our project is the development of computer algorithms that allow to describe stochastic systems when Gaussian approximations are not applicable. We concentrate primarily on rare-event situations. In these cases, Gaussian approximations are typically (but not always) inappropriate. We combine the insight in the asymptotic analysis that we perform to obtain algorithms that can be shown to run very fast for simulating rare events. The algorithms are shown to be optimal in a precise mathematical sense. For instance, we are able to compute probabilities that a phone call in a large call center (think of the 911 call center) gets dropped. Surely nobody would like to get his / her phone call unanswered in a life treatening situation. One of our papers discusses optimal algorithms for computing the probability that a call gets lost and the conditional distribution of the remaining times in the current calls in progress in a large call center. We specially consider situations such as the 911 call centers, that is, situations in which the loss probability is very small. It turn out that the methods developed here are also applicable to large insurance portfolios. These types of applications are still in their early stages, but we have in mind situations such as large pension fund systems (private or public), their modeling and their efficient design, specially in the presence of catastrophic events. Finally, we briefly mention that our research also covers spatial modeling with potential applications to environmental sciences. In this case we concentrate on extremes of Gaussian processes. We recognize that Gaussian assumptions are often unrealistic but sometimes models with non-Gaussian features are obtained out of non-linear transformations of Gaussian systems. The point is: Gaussian models are specially well suited for spatial dependence. An example to have in mind is the logarithm of the concentration of a contaminant in a city. One of our papers develops fast computational algorithms for the analysis of the contamination level in a whole area conditional on a sub-area with high contamination. We envision that these types of algorithms have can find applications, as we mentioned earlier, in environmental engineering.
