The Spectral Representation of Bessel Processes with Constant Drift: Applications in Queueing and Finance

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 λ0 = 0, followed by an infinite series of terms corresponding to the higher eigenvalues λn, n = 1, 2, . . . , as well as an integral over the continuous spectrum above μ2/2. When −1 < ν < − 1 2 and μ < 0, there is only one eigenvalue λ0 = 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.