Stochastic Models for Queueing and Finance

排队与融资的随机模型研究[j]。首先是对繁忙交通条件下具有截止日期的排队系统的分析。考虑带有更新流程到达流的排队系统。假设到达后，每个客户都被分配了一个前置时间，即客户获得服务的截止日期之前的时间。人们可以将车站排队的顾客的交货时间建模为实际线路上的计数措施，即与顾客交货时间相对应的点质量的位置。研究将解决这些测量值过程在大流量扩展下的收敛问题。研究也将导向金融的数学模型。在这样的模型上是一个交易账户的期权。对于其中最简单的一种，一个账户交易一个潜在的几何布朗运动和一个恒定利率的货币市场，有一个非常简单的最佳规则：当账户价值为负时持有几何布朗运动，当账户价值为正时持有几何布朗运动。对于两种几何布朗运动的一种期权，存在一种推测的最优规则，并得到数值分析的支持。这种证明似乎需要发展新的数学。第二个数学金融问题涉及信用衍生品风险中性定价统一模型的发展。本研究分为两部分。第一部分的主题，有时限的排队系统，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。数据延迟太长会导致不可接受的信号中断。拟议的研究将为考虑最后期限的重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，建立在由布莱克-斯科尔斯期权定价公式引发的金融革命的基础上。这里提出的特别工作涉及财产定价和金融工具的使用，其目的是防止损失，无论是由于市场价值的急剧减少（交易账户上的期权）还是违约（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).