Stochastic Models for Queueing and Finance

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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).