Stochastic Models for Queueing and Finance

排队和融资的随机模型提案：DMS-0103814主要研究者：Steven E. ShreveABSTRACT工作建议在两个领域。 第一个是在繁忙的交通条件下，分析了具有截止期的排队系统。 考虑具有更新过程到达流的排队系统。 假设在到达时，每个客户都被分配了一个提前期，即在客户的服务截止日期之前的时间。 人们可以把顾客在车站排队的提前期模型化为一种实际的计数方法，点群的位置对应于顾客的提前期。 研究将解决这些measure-valuedprocesses下的重流量缩放收敛。 研究也将针对金融数学模型。 在这样的模型是一个选择在一个交易帐户。 对于其中最简单的一个账户，即交易几何布朗运动和固定利率货币市场的账户，有一个非常简单的最优规则：当账户价值为负时，持有几何布朗运动多头，当账户价值为正时，持有几何布朗运动空头。 对于两个几何布朗运动上的期权，存在一个明确的最优规则，并得到数值分析的支持。 这个证明似乎需要新数学的发展。 第二个数学金融问题是关于信用衍生品风险中性定价的统一模型的发展。 第一部分的主题，具有截止期的交换系统，出现在通信网络中，特别是用于传输数字化视频或音频信号的网络中。 太长时间的数据延迟会导致信号的不可接受的中断.所提出的研究将为考虑截止日期的重负载通信网络的性能分析提供基础.第二部分的主题，金融数学模型，建立在由Black-Scholes期权定价公式开始的金融革命的基础上。 这里提出的具体工作是关于金融工具的正确定价和使用，其目的是保险损失，无论是由于市场价值的急剧下降（期权交易账户）或违约（信用衍生品）。

STOCHASTIC MODELS FOR QUEUEING AND FINANCENSF Proposal: DMS-0103814Principal Investigator: Steven E. ShreveABSTRACTWork is proposed in two areas. The first is the analysis of queueingsystems with deadlines under heavy traffic conditions. Consider aqueueing system with renewal process arrival streams. Suppose thatupon arrival, each customer is assigned a lead time, the amount of timeuntil the customer's deadline for service elapses. One can model the leadtimes of the customers in queue at a station as a counting measure on thereal line, the location of the point masses corresponding to customer leadtimes. Research will address the convergence of these measure-valuedprocesses under heavy-traffic scaling. Research will also be directed tomathematical models for finance. On such model is for an option on atraded account. For the simplest of these, an account trading oneunderlying geometric Brownian motion and a constant-interest-rate moneymarket, there is a strikingly simple optimal rule: hold the geometricBrownian motion long when the account value is negative and short whenthe account value is positive. For an option on two geometric Brownianmotions, there is a conjectured optimal rule, which is supported bynumerical analysis. The proof appears to require the development of newmathematics. A second mathematical finance problem concerns thedevelopment of a unifying model for risk-neutral pricing of creditderivatives.This research has two parts. The subject of the first part, queueingsystems with deadlines, arise in communication networks, especiallynetworks used to transmit digitized video or audio signals. Data whichare too long delayed can cause unacceptable disruption of the signal.The proposed research will provide a basis for performance analysis ofheavily-loaded communication networks which take deadlines into account.The subject of the second part, mathematical models for finance, builds onthe revolution in finance begun by the Black-Scholes option pricing formula. The particular work proposed here is concerned with properpricing and usage of financial instruments whose purpose is to insureagainst loss, either due to drastic reduction in market value (options ona traded account) or default (credit derivatives).