Logarithmic Lévy process directed by Poisson subordinator
Volume 6, Issue 4 (2019), pp. 419–441
Pub. online: 4 October 2019
Type: Research Article
Open Access
Open Access
Received
5 April 2019
5 April 2019
Revised
13 September 2019
13 September 2019
Accepted
13 September 2019
13 September 2019
Published
4 October 2019
4 October 2019
Abstract
Let $\{L(t),t\ge 0\}$ be a Lévy process with representative random variable $L(1)$ defined by the infinitely divisible logarithmic series distribution. We study here the transition probability and Lévy 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 Lévy 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 Lévy measure of the subordinated process $Z(t)$ is a shifted Lévy measure of the Negative-Binomial process $X(t)$. We compare the properties of these processes, knowing that the total masses of corresponding Lévy measures are equal.
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