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The spectral representation of Bessel processes with constant drift: applications in queueing and finance

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Vadim Linetsky
Vadim Linetsky*
Affiliation:
Northwestern University
*
∗ Postal address: Department of Industrial Engineering and Management Sciences, McCormick School of Engineering and Applied Sciences, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA. Email address: linetsky@iems.nwu.edu
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Abstract

Bessel processes with constant negative drift have recently appeared as heavy-traffic limits in queueing theory. We derive a closed-form expression for the spectral representation of the transition density of the Bessel process of order ν > −1 with constant drift μ ≠ 0. When ν > -½ and μ < 0, the first term of the spectral expansion is the steady-state gamma density corresponding to the zero principal eigenvalue λ 0 = 0, followed by an infinite series of terms corresponding to the higher eigenvalues λ n , n = 1,2,…, as well as an integral over the continuous spectrum above μ 2/2. When −1 < ν < -½ and μ < 0, there is only one eigenvalue λ 0 = 0 in addition to the continuous spectrum. As well as applications in queueing, Bessel processes with constant negative drift naturally lead to two new nonaffine analytically tractable specifications for short-term interest rates, credit spreads, and stochastic volatility in finance. The two processes serve as alternatives to the CIR process for modelling mean-reverting positive economic variables and have nonlinear infinitesimal drift and variance. On a historical note, the Sturm–Liouville equation associated with Bessel processes with constant negative drift is closely related to the celebrated Schrödinger equation with Coulomb potential used to describe the hydrogen atom in quantum mechanics. Another connection is with D. G. Kendall's pole-seeking Brownian motion.

Keywords

Bessel processpole-seeking Brownian motionCoulomb potentialspectral expansionheavy traffic limitCIR modelmodelinterest-rate model

MSC classification

Primary: 60J35: Transition functions, generators and resolvents 60J60: Diffusion processes 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 327 - 344
Copyright
Copyright © Applied Probability Trust 2004 

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