Logarithmic L\'{e}vy process directed by Poisson subordinator

TL;DR

This paper investigates a logarithmic Lévy process driven by a Poisson subordinator, analyzing its transition probabilities, Lévy measure, and related subordinated processes, revealing new properties involving iterated logarithmic functions.

Contribution

It introduces and studies a logarithmic Lévy process directed by a Poisson process, including its transition probability, Lévy measure, and relationships with subordinated processes, highlighting novel properties.

Findings

Lévy measure of the process is explicitly characterized.

The Bernstein functions involve iterated logarithmic functions.

Lévy measure of the subordinated process is a shifted version.

Abstract

Let $\{L(t),t\geq 0\}$ be a L\'{e}vy process with representative random variable $L(1)$ defined by the infinitely divisible logarithmic series distribution. We study here the transition probability and L\'{e}vy measure of this process. We also define two subordinated processes. The first one, $Y(t)$, is a Negative-Binomial process $X(t)$ directed by Gamma process. The second process, $Z(t)$, is a Logarithmic L\'{e}vy process $L(t)$ directed by Poisson process. For them, we prove that the Bernstein functions of the processes $L(t)$ and $Y(t)$ contain the iterated logarithmic function. In addition, the L\'{e}vy measure of the subordinated process $Z(t)$ is a shifted L\'{e}vy measure of the Negative-Binomial process $X(t)$. We compare the properties of these processes, knowing that the total masses of corresponding L\'{e}vy measures are equal.