A Class of Probability Distributions that is Closed with Respect to Addition as Well as Multiplication of Independent Random Variables

Structural properties of generalised Planck distributions

A family of generalised Planck (GP) laws is defined and its structural properties explored. Sometimes subject to parameter restrictions, a GP law is a randomly scaled gamma law; it arises as the equilibrium law of a perturbed version of the Feller mean reverting diffusion; the density functions can be decreasing, unimodal or bimodal; it is infinitely divisible. It is argued that the GP law is not a generalised gamma convolution. Characterisations are obtained in terms of invariance under random contraction of a weighted version of a related law. The GP law is a particular instance of equilibrium laws obtained from a recursion suggested by a genetic mutation-selection balance model. Some related infinitely divisible laws are exhibited.

On the law of homogeneous stable functionals

LetAbe theLq-functional of a stable Lévy process starting from one and killed when crossing zero. We observe thatAcan be represented as the independent quotient of two infinite products of renormalized Beta random variables. The proof relies on Markovian time change, the Lamperti transformation, and an explicit computation performed in [38] on perpetuities of hypergeometric Lévy processes. This representation allows us to retrieve several factorizations previously shown by various authors, and also to derive new ones. We emphasize the connections betweenAand more standard positive random variables. We also investigate the law of Riemannian integrals of stable subordinators. Finally, we derive several distributional properties ofArelated to infinite divisibility, self-decomposability, and the generalized Gamma convolution.

Windings of planar processes, exponential functionals and Asian options

Motivated by a common mathematical finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of Brownian motion in the framework of planar Brownian motion. We prove a conjecture of Vakeroudis and Yor (2012) concerning infinite divisibility properties of this random variable and present a novel simple proof of the result of DeBlassie (1987), (1988) concerning the asymptotic behavior of the distribution of the Bessel clock appearing in the skew-product representation of planar Brownian motion, as t→∞. We use the results of the windings approach in order to obtain results for quantities associated to the pricing of Asian options.