Two models of default from finance and a model of invasion from ecology

My thesis consists of three different projects. 1) Investors are exposed to credit risk due to the possibility that one or more counterparties in a financial agreement will default; that is, not honor their obligations to make certain payments. It is usually not enough to consider the default of a single firm because of the effect of contagion—the default of one firm is dependent of the other firms in the economy. This project considers static models of default that have appeared in the mathematical finance literature. These models are constructed from an underlying graph with a set of nodes V representing firms. They give a probability distribution on {0, 1}V, where a 1 in the k th coordinate indicates that the kth firm has defaulted at the end of a particular time period. The drawback of these models is that they are static—they do not try to say anything about the distribution of the default times of a group of firms. It is therefore of interest to try to give these models Markovian dynamics. In Chapter 1 we show in several natural cases that this is not possible.;2) In ecology, the extinction of a population can be described as the first passage through some threshold value for the diffusion process which represents the number of individuals. Similarly, in finance, the default time of a counterparty is sometimes modeled as the first passage time of a credit index process below a barrier. It is therefore relevant to consider the following question: If we know the distribution of the default time can we find a unique barrier which gives this distribution? This is known as the Inverse First Passage Time (IFPT) problem in the literature. We consider a more general 'smoothed' version of the inverse first passage time problem in which the first passage time is replaced by the first instant that the time spent below the barrier exceeds an independent exponential random variable. In Chapter 2 we show that any smooth distribution results from some unique continuously differentiable barrier.;3) A fundamental problem in ecology is to understand when it is possible for one species to invade the range of another, established species. Mathematical models for invasibility have contributed significantly to the understanding of the epidemiology of infectious disease outbreaks ([CLSJG05]) and ecological processes ([LM96], [Cas01]).;There is widespread empirical evidence that invasions can occur when there is significant heterogeneity in space and time in the range of the resident species. This heterogeneity can arise due to variability in abiotic factors (e.g. precipitation, temperature or sunlight) or biotic factors (presence of other competitors or predators). There have only been a a few studies that try to explain how spatio-temporal heterogeneity facilitates invasibility (see, for example, [SLS09]).;Using ideas from [ERSS], we propose in Chapter 3 a general model of the invasion process with a view to understanding what factors make invasion possible. We consider a stochastic differential equation (SDE) model of a resident population that is living in an environment consisting of n patches and is subject to an attempted invasion by another species.