Thorin processes and their subordination
Lorenzo Torricelli
TL;DR
This paper develops a comprehensive theory of Thorin processes, a structured subclass of infinitely divisible processes whose one-time laws lie in the Thorin class $T(\mathbb{R}^d)$. It establishes a Thorin–Bondesson representation that clarifies the role of the Thorin measure in determining moments, exponential moments, density regularity, and path variation, and it links these properties to the Lévy framework via a polar decomposition. A central contribution is a detailed subordination theory: subordinating a Lévy or Brownian process by a Thorin subordinator yields Thorin processes with explicitly computable Thorin triplets, and conversely, certain Thorin processes admit Brownian-subordinator representations through explicit pushforwards of the Thorin measure. The paper also catalogues representative Thorin distributions (multivariate Gamma, stable, tempered stable, logistic-type) and analyzes their exponential moments, tails, and convolution-equivalence properties, aided by examples and counterexamples such as the Dickman distribution. Overall, the Thorin perspective provides tractable formulae for subordination, density analysis, and moment calculations, enhancing both theory and potential applications in finance and risk modeling where Gamma convolutions arise.
Abstract
A Thorin process is a stochastic process with independent and stationary increments whose laws are weak limits of finite convolutions of gamma distributions. Many popular Lévy processes fall under this class. The Thorin class can be characterized by a representing triplet that conveys more information on the process compared to the Lévy triplet. In this paper we investigate some relationships between the Thorin structure and the process properties, and find that the support of the Thorin measure characterizes the existence of the critical exponential moment, as well as the asymptotic equivalence between the Lévy tail function and the complementary distribution function. Furthermore, it is illustrated how univariate Brownian subordination with respect to Thorin subordinators produces Thorin processes whose representing measure is given by a pushforward with respect to a hyperbolic function, leading to arguably easier formulae compared to the Bochner integral determining the Lévy measure. We provide a full account of the theory of multivariate Thorin processes, starting from the Thorin--Bondesson representation for the characteristic exponent, and highlight the roles of the Thorin measure in the analysis of density functions, moments, path variation and subordination. Various old and new examples are discussed. We finally detail a treatment of subordination of gamma processes with respect to negative binomial subordinators which is made possible by the Thorin-Bondesson representation.