Density of generalized Verhulst process and Bessel process with constant drift

Abstract

In this paper, we derive exact closed-form density functions of the generalized Verhulst process and the Bessel process with constant drift, which have applications in mathematical biology and queueing theory. We propose a generic probabilistic method for deriving exact closed-form density functions for these two diffusion processes based on a novel application of the exponential measure change in [T. Hurd and A. Kuznetsov, Explicit formulas for Laplace transforms of stochastic integrals, Markov Process. Relat. Fields, 14(2):277–290, 2008], together with formulae in [A. Borodin and P. Salminen, Handbook of Brownian Motion – Facts and Formulae, Birkhauser, Basel, 2015]. Our study generalizes several known results in the literature.