On exponential functionals, harmonic potential measures and undershoots of subordinators

Abstract

We establish a link between the distribution of an exponential functional I and the undershoots of a subordinator, which is given in terms of the associated harmonic potential measure. This allows us to give a necessary and sufficient condition in terms of the Lévy measure for the exponential functional to be multiplicative infinitely divisible. We then provide a formula for the moment generating function of an exponential functional $I$ and the so called remainder random variable $R$ associated to it. We provide a realization of the remainder random variable $R$ as an infinite product involving independent last position random variables of the subordinator. Some properties of harmonic measures are obtained and some examples are provided.

24 pages

References in corpus (2)

• Asymptotic laws for compositions derived from transformed subordinators
• On the Mellin transforms of the perpetuity and the remainder variables associated to a subordinator

Cited by in corpus (4)

• A note on a Poissonian functional and a $q$-deformed Dufresne identity
• Complete monotonicity and bernstein properties of functions are characterized by their restriction on N
• On a moment problem related to Bernstein functions
• Complete Bernstein functions and subordinators with nested ranges. A note on a paper by P. Marchal