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Multivariate subordinators are multivariate Lévy processes that are increasing in each component. Various examples of multivariate subordinators, of interest for applications, are given. Subordination of Lévy processes with independent components by multivariate subordinators is defined. Multiparameter Lévy processes and their subordination are introduced so that the subordinated processes are multivariate Lévy processes. The relations between the characteristic triplets involved are established. It is shown that operator self-decomposability and the operator version of the class Lm property are inherited from the multivariate subordinator to the subordinated process under the condition of operator stability of the subordinand.

Let R + N = [ 0 , ∞ ) N . We here make new contributions concerning a class of random fields ( X t ) t ∈ R + N which are known as multiparameter Lévy processes. Related multiparameter semigroups of operators and their generators are represented as pseudo-differential operators. We also provide a Phillips formula concerning the composition of ( X t ) t ∈ R + N by means of subordinator fields. We finally define the composition of ( X t ) t ∈ R + N by means of the so-called inverse random fields, which gives rise to interesting long-range dependence properties. As a byproduct of our analysis, we present a model of anomalous diffusion in an anisotropic medium which extends the one treated in Beghin et al. (Stoch Proc Appl 130:6364–6387, 2020), by improving some of its shortcomings.

Lévy bandits are multi-armed bandits driven by Lévy processes. As anticipated from existing research, Lévy bandits are optimally controlled by an index strategy: One can associate with each arm an index function of its state, and optimal strategies are those that allocate time to arms whose states have the largest index. Furthermore, the index function of an arm is calculated independently of the other arms, and the optimal reward can be expressed in terms of the indices. Somewhat less anticipated, however, is the fact that the index function of an arm, driven by a Lévy process, has a representation in terms of the decreasing ladder sets and the exit system of its Lévy driver. Moreover, the Wiener-Hopf factorization of the Lévy exponents of an arm can be used to obtain the characteristic function of some excursion law, through which the index of the arm is defined. We use this factorization to calculate explicitly index functions and optimal rewards of some interesting Lévy bandits, rediscovering along the way that local time naturally quantifies switching in continuous time.