## 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 language | English (US) |
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Pages (from-to) | 327-344 |

Number of pages | 18 |

Journal | Journal of Applied Probability |

Volume | 41 |

Issue number | 2 |

DOIs | |

State | Published - 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