Mathematical Finance and Stochastic Networks

这个建议有两个部分。关于随机网络的部分将考虑当任务有截止日期时在繁忙交通中的队列网络。这些任务的交付时间（截止日期前的时间）被建模为实线上的计数度量。作为先前项目的一部分，当网络流量强度接近1时，确定了这些测量值过程的极限。本项目将确定极限测量值过程与极限前过程之间的差异，以便确定使用极限过程作为重载网络模型的准确性。该提案的第二部分处理金融市场中的信用风险。特别令人感兴趣的是贷款抵押债券的价差变动。这些响应市场对构成结构的贷款违约概率的预期，但以一种高度非线性的方式。计算机网络、制造网络和电话网络有一个共同的特点，即任务（例如，消息、硅片、电话）在随机时间到达站点（例如，计算机、机器、交换机），并且需要随机数量的服务。在大流量条件下，这些网络的性能可以用布朗运动理论来分析。在这个项目中，我们考虑任务到达并且必须在截止日期之前通过和退出网络的网络。该项目将开发一种方法来确定网络将在截止日期内处理的任务比例。布朗运动也是建立金融市场价格行为模型的基本过程。该提案的第二部分将考虑由贷款支付支持的金融资产的价格（例如，抵押贷款支持证券）。这些资产对支持它们的贷款的信用风险很敏感。为这些资产的价格变动建立可靠的模型是衡量和控制与持有和交易这些证券相关的风险的重要一步。

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.