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 (see Mackevicius (2015), Jakubowski and Wisniewolski (2015)), and the Bessel process with a constant drift (see Coman et al (1998), Linetsky (2004)), 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 diusion processes based on a novel application of the exponential measure change in Palmowski and Rolski (2002) and Hurd and Kuznetsov (2008), together with formulae in Borodin and Salminen (2015). Our study generalizes several known results in the literature: Proposition 2.1 generalizes results of Theorem 2 and Theorem 3 in Jakubowski and Wisniewolski (2015) to the case of the generalized Verhulst process. Proposition 2.3 provides the exact closed-form density for a Bessel process with a constant drift, and it provides an alternative expression to the spectral expansion given in Proposition 1 of Linetsky (2004).