Stochastic Models for Queueing and Finance

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

STOCHASTIC MODELS FOR QUEUEING AND FINANCE NSF Proposal: DMS-0103814 Principal Investigator: Steven E. Shreve ABSTRACT Work is proposed in two areas. The first is the analysis of queueing systems with deadlines under heavy traffic conditions. Consider a queueing system with renewal process arrival streams. Suppose that upon arrival, each customer is assigned a lead time, the amount of time until the customer's deadline for service elapses. One can model the lead times of the customers in queue at a station as a counting measure on the real line, the location of the point masses corresponding to customer lead times. Research will address the convergence of these measure-valued processes under heavy-traffic scaling. Research will also be directed to mathematical models for finance. On such model is for an option on a traded account. For the simplest of these, an account trading one underlying geometric Brownian motion and a constant-interest-rate money market, there is a strikingly simple optimal rule: hold the geometric Brownian motion long when the account value is negative and short when the account value is positive. For an option on two geometric Brownian motions, there is a conjectured optimal rule, which is supported by numerical analysis. The proof appears to require the development of new mathematics. A second mathematical finance problem concerns the development of a unifying model for risk-neutral pricing of credit derivatives. This research has two parts. The subject of the first part, queueing systems with deadlines, arise in communication networks, especially networks used to transmit digitized video or audio signals. Data which are too long delayed can cause unacceptable disruption of the signal. The proposed research will provide a basis for performance analysis of heavily-loaded communication networks which take deadlines into account. The subject of the second part, mathematical models for finance, builds on the revolution in finance begun by the Black-Scholes option pricing formula. The particular work proposed here is concerned with proper pricing and usage of financial instruments whose purpose is to insure against loss, either due to drastic reduction in market value (options on a traded account) or default (credit derivatives).