The average of a negative-binomial Lévy process and a class of Lerch distributions

Abstract

In this paper we discuss the average of a Lévy process with a marginal negative-binomial distribution taken over a finite time interval, and simultaneously introduce a new class of absolutely continuous distribution based on Lerch’s transcendent. Various distribution formulas are obtained in explicit form, including characteristic functions, distribution functions and moments. Some interesting asymptotics are also analyzed. As a consequence, we obtain rapidly converging series representations for the probability distribution of the average process. Numerical examples are provided in order to illustrate the proposed formulas.

Keywords:

• Average process
• characteristic functions
• distribution functions
• moments
• negative-binomial distribution
• Lerch’s transcendent

MSC 2010 CLASSIFICATION:

• 60E10
• 60G51
• 60G55

Notes

1 It is automatically understood that $B_0=0$ a.s., and B has independent stationary increments as well as a.s. càdlàg trajectories.

2 Of order 2, ${\text{Li}}_2(·)$ is specifically referred to as dilogarithm. See (Abramowitz and Stegun 1972, $\S$27.7).

3 In this paper we describe density functions, wherever finite, as being càdlàg in the presence of discontinuities.

4 Lerch’s transcendent is closely related to the Hurwitz-Lerch zeta function that generalizes the well-known Hurwitz zeta function. See, e.g., (Bateman and Erdélyi 1953, $\S$1.11) or (Gradshteyn and Ryzhik 2007, $\S$9.55).

5 The jumps are said to be downward in terms of increasing x.