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The concept of a L\'evy subordinator is generalized to a family ofnon-decreasing stochastic processes, which are parameterized in terms of twoBernstein functions. Whereas the independent increments property is onlymaintained in the L\'evy subordinator special case, the considered family isalways strongly infinitely divisible with respect to time, meaning that a pathcan be represented in distribution as a finite sum with arbitrarily manysummands of independent and identically distributed paths of another process.Besides distributional properties of the process, we present two applicationsto the design of accurate and efficient simulation algorithms. First, eachmember of the considered family corresponds uniquely to an exchangeablemax-stable sequence of random variables, and we demonstrate how the associatedextreme-value copula can be simulated exactly and efficiently from its Pickandsdependence measure. Second, we show how one obtains different series andintegral representations for infinitely divisible probability laws by varyingthe parameterizing pair of Bernstein functions, without changing the law ofone-dimensional margins of the process. As a particular example, we present anexact simulation algorithm for compound Poisson distributions from theBondesson class, for which the generalized inverse of the distribution functionof the associated Stieltjes measure can be evaluated accurately.

alea

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