Subordinators which are infinitely divisible w.r.t. time: Construction, properties, and simulation of max-stable sequences and infinitely divisible laws

TL;DR

This paper generalizes Lévy subordinators to a broader family of non-decreasing processes parameterized by Bernstein functions, exploring their properties, simulation methods, and applications to max-stable sequences and infinitely divisible laws.

Contribution

It introduces a new class of infinitely divisible processes with strong time-divisibility, extending Lévy subordinators, and provides novel simulation algorithms for related max-stable sequences and probability laws.

Findings

The family is strongly infinitely divisible with respect to time.

Exact simulation of exchangeable max-stable sequences is achieved.

Series and integral representations for laws are obtained by parameter variation.

Abstract

The concept of a L\'evy subordinator is generalized to a family of non-decreasing stochastic processes, which are parameterized in terms of two Bernstein functions. Whereas the independent increments property is only maintained in the L\'evy subordinator special case, the considered family is always strongly infinitely divisible with respect to time, meaning that a path can be represented in distribution as a finite sum with arbitrarily many summands of independent and identically distributed paths of another process. Besides distributional properties of the process, we present two applications to the design of accurate and efficient simulation algorithms. First, each member of the considered family corresponds uniquely to an exchangeable max-stable sequence of random variables, and we demonstrate how the associated extreme-value copula can be simulated exactly and efficiently from its Pickands dependence measure. Second, we show how one obtains different series and integral representations for infinitely divisible probability laws by varying the parameterizing pair of Bernstein functions, without changing the law of one-dimensional margins of the process. As a particular example, we present an exact simulation algorithm for compound Poisson distributions from the Bondesson class, for which the generalized inverse of the distribution function of the associated Stieltjes measure can be evaluated accurately.