The spectral representation of bessel processes with constant drift: Applications in queueing and finance

Vadim Linetsky*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

26 Scopus citations

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 > -1/2 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 < -1/2 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.

Original languageEnglish (US)
Pages (from-to)327-344
Number of pages18
JournalJournal of Applied Probability
Volume41
Issue number2
DOIs
StatePublished - Jun 2004

Keywords

  • 3/2 model
  • Bessel process
  • CIR model
  • Coulomb potential
  • Heavy traffic limit
  • Interest-rate model
  • Pole-seeking Brownian motion
  • Spectral expansion

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Statistics, Probability and Uncertainty

Fingerprint Dive into the research topics of 'The spectral representation of bessel processes with constant drift: Applications in queueing and finance'. Together they form a unique fingerprint.

Cite this