subordinated Lévy processes Poisson random measure pure jump Markov processes Kolmogorov backward equation
Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 40, No 3-4, (2014), 183p-208p
We consider the jump structure of the subordinated Lévy processes and subordinated Markov branching processes. Subordination provides a method of constructing a large subclass of Markov or Lévy processes Y (t) = X(T (t)), where X(t) is a Markov or Lévy process and T (t) is a continuous time subordinator independent of X(t); that is a Lévy process with positive increments and T (0) = 0. Let X(t) be a Lévy process. Then subordination preserves the independence and stationarity of the increments, but it changes their amplitudes and the total mass of the Lévy measure. Let X(t) be a Markov branching process. Then subordination (owing to the independence of X(t) and T (t)) preserves the Markov property, but it disturbs the branching property. The infinitesimal generator of the subordinated process Y (t) involves the total progeny of reproduction. The intensity of the jump times depends on the subordinator’s Bernstein function. 2010 Mathematics Subject Classification: 60J80, 60K05.