Mathematical Finance and Stochastic Networks

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This proposal has two parts. The part on stochastic networks will consider networks of queues in heavy traffic when tasks have due dates. The lead times (time until due date) of these tasks are modeled as counting measures on the real line. As part of a previous project, the limit of these measure-valued processes was identified as the network traffic intensity approached one. This project will identify the difference between the limiting measure-valued process and the pre-limit processes, so that the accuracy of using the limiting process as a model for a heavily loaded network can be determined. The second part of the proposal treats credit risk in financial markets. Of particular interest is the spread movements of tranches of collaterialized loan obligations. These respond to market expectations concerning the default probabilities of the loans composing the structure, but in a highly nonlinear way.Computer networks, manufacturing networks and telephone networks have the common feature that tasks (e.g., messages, silicon wafers, telephone calls) arrive at stations (e.g., computers,, machines, switches) at random times and require random amounts of service. Under heavy traffic conditions, the performance of these networks can be analyzed using the theory of Brownian motion. In this project, we consider networks in which tasks arrive and must move through and exit the network ahead of deadlines. This project will develop a method to determine what proportion of tasks a network will process within their deadlines. Brownian motion is also the fundamental process for building models of the behavior of prices in financial markets. A second part of this proposal will consider prices of financial assets that are backed by loan payments (e.g., mortgage-backed securities). These assets are sensitive to the credit risk of the loans backing them. Construction of reliable models for the price movements of these assets is an important step in measuring and controlling the risk associated with holding and trading these securities.