Lévy measures of infinitely divisible positive processes: examples and distributional identities (Q6150880)

A random process is infinitely divisible if all its finite dimensional marginals are infinitely divisible. The infinite divisibility is characterized by the existence of a unique Lévy measure. The existence and uniqueness of such measure is well-known. It might be difficult to obtain an expression for the Lévy measure directly from the representation of characteristic functions. However, there were established some relations between characteristic function, some auxiliary process and Lévy measure. To illustrate these relations, the authors consider simple examples of nonnegative infinitely divisible processes. In each case the Lévy measure is directly computable from the representation of characteristic functions or from the stochastic integral representation of the process. The authors present remarkable identities satisfied by the considered nonnegative infinitely divisible processes. Moreover, the general expression for the Lévy measure provides alternative formulas for this measure, which are also remarkable. The authors treat the cases of Poisson processes, Sato processes, stochastic convolutions, and tempered stable subordinators. They also point out a connection with infinitely divisible random measures. The case of infinitely divisible permanental processes is treated, which is the first case for which identities in law related to auxiliary process have been established. In this case, such identities in law are called ``isomorphism theorems'' in reference to the first one established by Dynkin, the so-called ``Dynkin isomorphism theorem''. For the entire collection see [Zbl 07730209].